Math Kangaroo http://www.mathkangaroo.org/2010page/Clark/clark/pdb/#Past Contests
Grade 01-02 Problem Kangur_2005_02_1 (3 pts) http://www.mathkangaroo.org In the enchanted garden of the Green King, there are apple trees that produce golden apples. Every day, 5 golden apples become ripe on each tree, and at the end of each day they fall from the trees. Today, the Green Gardener has picked up 20 ripe apples that fell under the trees last night. How many enchanted trees are there in the garden? A) 4 B) 5 C) 6 D) 7 E) 8 Problem Kangur_2005_02_2 (3 pts) http://www.mathkangaroo.org Alma, Maria, Anne and Michael had 2 apples each. Each one ate one apple. How many apples do they now have altogether? A) 1 B) 2 C) 4 D) 6 E) 8 Problem Kangur_2005_02_3 (3 pts) http://www.mathkangaroo.org Only one digit from 1 to 9 is repeated three times in this drawing.
The rest of the digits are repeated twice. Which digit is repeated three times? A) 9 B) 8 C) 3 D) 4 E) 7
Problem Kangur_2005_02_4 (3 pts) http://www.mathkangaroo.org How many different digits can you see in the picture below?
A) 3 B) 4 C) 5 D) 6 E) 9 Problem Kangur_2005_02_5 (3 pts) http://www.mathkangaroo.org What number is hidden under the question mark in the picture below (on the last car)?
A) 7 B) 4 C) 0 D) 1 E) 2 Problem Kangur_2005_02_6 (3 pts) http://www.mathkangaroo.org 5-4+3-2+1=? A) 1 B) 2 C) 3 D) 4 E) 0 Problem Kangur_2005_02_7 (3 pts) http://www.mathkangaroo.org When Ann was born, Michael was 4. Now Ann is 3 years old. How old is Michael? A) 1 B) 6 C) 7
D) 8 E) 10 Problem Kangur_2005_02_8 (3 pts) http://www.mathkangaroo.org How many blocks were used to build the figure shown in the picture?
A) 7 B) 12 C) 13 D) 14 E) 16 Problem Kangur_2005_02_9 (4 pts) http://www.mathkangaroo.org Which four beads below need to be added to this string:
Problem Kangur_2005_02_10 (4 pts) http://www.mathkangaroo.org How many more square tiles do we need to put on the kitchen floor to cover all of it? (See the picture.)
A) 12 B) 10 C) 9 D) 6 E) 4 Problem Kangur_2005_02_11 (4 pts) http://www.mathkangaroo.org Helga is climbing stairs in such a way that she goes up 2 steps at a time. She is standing on the third step now. On which step will she be after she moves up 3 times? A) 9 B) 1 C) 6
D) 5 E) 8 Problem Kangur_2005_02_12 (4 pts) http://www.mathkangaroo.org Some pages are missing from an open book. On the left page you can see page number 12 and on the right page you can see page number 15. How many pages are missing? A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2005_02_13 (4 pts) http://www.mathkangaroo.org One hen lays one egg a day. In how many days will two hens lay 6 eggs? A) 1 B) 3 C) 6 D) 9 E) 10 Problem Kangur_2005_02_14 (4 pts) http://www.mathkangaroo.org There are two horses, one duck, one fish, an eagle, and a boy in a private garden. How many legs do they have altogether? A) 10 B) 12 C) 14 D) 16 E) 18 Problem Kangur_2005_02_15 (4 pts) http://www.mathkangaroo.org Anne has some apples. Maria has 2 apples more than Anne. Altogether they have 8 apples. How many apples does Anne have? A) 3 B) 5 C) 6 D) 7 E) 10 Problem Kangur_2005_02_16 (4 pts) http://www.mathkangaroo.org
Which figure is next in this sequence:
Problem Kangur_2005_02_17 (5 pts) http://www.mathkangaroo.org Monika is 2 years old and Karl is 4. How old will Monika be when Karl is 11? A) 6 B) 7 C) 8 D) 9 E) 10 Problem Kangur_2005_02_18 (5 pts) http://www.mathkangaroo.org During the race, right before the finish line, I passed the runner who won the third place. What place did I win? A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2005_02_19 (5 pts) http://www.mathkangaroo.org There are three weights on the scales in the left picture: 1kg, 4 kg, 2 kg and just one weight of 1 kg on the scales in the right picture. What is the weight of the fruit in the basket?
A) 7 kg B) 6 kg C) 5 kg D) 3 kg E) 2 kg Problem Kangur_2005_02_20 (5 pts) http://www.mathkangaroo.org Which set of signs +, - needs to be used to make this expression true?
A) +, -, + B) -, +, C) +, +, D) +, +, + E) -, +, + Problem Kangur_2005_02_21 (5 pts) http://www.mathkangaroo.org The sum of two digits, one from inside the square and one from outside the square is greater than 10. How many such pairs can you make?
A) 19 B) 11 C) 6 D) 24 E) 18 Problem Kangur_2005_02_22 (5 pts) http://www.mathkangaroo.org What number is covered by ? in the last picture below?
A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2005_02_23 (5 pts) http://www.mathkangaroo.org Hans fills the square table with numbers in such a way that the sum of the numbers in each column is 15 and the sum of the numbers in each row is 15 and the sum on each diagonal is 15. What number will he put in place of ? ?
A) 1 B) 3 C) 2
D) 6 E) 9 Problem Kangur_2005_02_24 (5 pts) http://www.mathkangaroo.org A train has four cars in four colors: red, green, white and yellow. The green car is not the first nor the last. The yellow car is not next to the white car nor next to the red car. The first car is white. What is the order of the cars in that train? A) white, green, red, yellow B) white, yellow, green, red C) green, yellow, red, white D) red, white, green, yellow E) white, red, green, yellow Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Grade 03-04 Problem Kangur_2005_0304_1 (3 pts) http://www.mathkangaroo.org A butterfly sat down on a correctly solved problem. What number did it cover up?
A) 250 B) 400 C) 500 D) 910 E) 1800 Problem Kangur_2005_0304_2 (3 pts) http://www.mathkangaroo.org At noon, the minute hand of a clock is in the following position:
What will the position of the minute hand be after 17 quarters of an hour?
A)
B)
C)
D)
E)
Problem Kangur_2005_0304_3 (3 pts) http://www.mathkangaroo.org Joan bought some cookies, each of which costs 3 dollars. She gave the salesperson 10 dollars, and received 1 dollar as change. How many cookies did Joan buy? A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2005_0304_4 (3 pts) http://www.mathkangaroo.org After the trainer's first whistle, the monkeys at the circus formed 4 rows. There were 4 monkeys in each row. After the second whistle, they rearranged themselves into 8 rows. How many monkeys were there in each row after the second whistle?
A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2005_0304_5 (3 pts) http://www.mathkangaroo.org Eva lives with her parents, her brother, one dog, two cats, two parrots, and four fish. What is the total number of legs that they have altogether? A) 22 B) 24 C) 28 D) 32 E) 40 Problem Kangur_2005_0304_6 (3 pts) http://www.mathkangaroo.org John has a chocolate bar consisting of square pieces 1 cm x 1 cm in size. He has already eaten some of the corner pieces (see the picture). How many pieces does John have left?
A) 66 B) 64 C) 62 D) 60 E) 58 Problem Kangur_2005_0304_7 (3 pts) http://www.mathkangaroo.org Two traffic signs mark the bridge in my village (see the picture below). These signs indicate the maximum vehicle width and the maximum vehicle weight allowed on the bridge. Which one of the following trucks is allowed to cross that bridge?
A) It is 315 cm wide and it weights 4400 kg. B) It is 330 cm wide and it weights 4250 kg. C) It is 325 cm wide and it weights 4400 kg.
D) It is 330 cm wide and it weights 4200 kg. E) It is 325 cm wide and it weights 4250 kg. Problem Kangur_2005_0304_8 (3 pts) http://www.mathkangaroo.org Each of seven boys has paid the same amount of money for a trip. The total sum of what they paid is a three digit number, which can be written as 3 0. What is the middle digit of this number? A) 3 B) 4 C) 5 D) 6 E) 7 Problem Kangur_2005_0304_9 (4 pts) http://www.mathkangaroo.org What is the smallest possible number of children in a family where each child has at least one brother and at least one sister? A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2005_0304_10 (4 pts) http://www.mathkangaroo.org Out of all five numbers below, I chose one. The number is even and all of its digits are different. The hundreds digit is double the ones digit. The tens digit is greater than the thousands digit. Which number did I choose? A) 1246 B) 3874 C) 4683 D) 4874 E) 8462 Problem Kangur_2005_0304_11 (4 pts) http://www.mathkangaroo.org A square piece of paper has been cut into three pieces. Two of them are shown in the picture:
Which of the pieces below is the third one?
A)
B)
C)
D)
E)
Problem Kangur_2005_0304_12 (4 pts) http://www.mathkangaroo.org An elevator cannot carry more than 150 kg. Four friends weigh 60 kg, 80 kg, 80 kg, and 80 kg, respectively. What is the least number of trips necessary to carry the four friends to the highest floor? A) 1 B) 2 C) 3 D) 4 E) 7 Problem Kangur_2005_0304_13 (4 pts) http://www.mathkangaroo.org Ala has 24 dollars, Barb has 66 dollars, and Sophia has as many dollars more than Ala as she has less than Barb. How many dollars does Sophia have? A) 33 B) 35 C) 42 D) 45 E) 48 Problem Kangur_2005_0304_14 (4 pts) http://www.mathkangaroo.org There are eight kangaroos in the cells of the table (see the picture). What is the least number of the kangaroos that need to be moved to the empty cells so that there would be exactly two kangaroos in any row and in any column of the table?
A) 4 B) 3 C) 2 D) 1 E) 0
Problem Kangur_2005_0304_15 (4 pts) http://www.mathkangaroo.org Greg needs to bring four full sacks of sand from the river to a house located at the other end of the village. Unfortunately, on his way through the village, half of the sand spills out of the sack through a hole. How many trips does Greg need to make from the river to the house in order to bring the required amount of sand? A) 4 B) 5 C) 6 D) 7 E) 8 Problem Kangur_2005_0304_16 (4 pts) http://www.mathkangaroo.org During a Kangaroo camp, Adam solved five problems every day, and Brad solved two problems daily. After how many days did Brad solve as many problems as Adam solved in 4 days? A) After 5 days B) After 7 days C) After 8 days D) After 10 days E) After 20 days Problems 5 points each Problem Kangur_2005_0304_17 (5 pts) http://www.mathkangaroo.org There were 9 pieces of paper. Some of them were cut into three pieces. As a result, there are 15 pieces of paper now. How many pieces of paper were cut? A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2005_0304_18 (5 pts) http://www.mathkangaroo.org Using 6 matches, only one rectangle with a perimeter of 6 matches can be made (see the picture). How many different rectangles with a perimeter of 14 matches can be made using 14 matches?
A) 2 B) 3 C) 4
D) 6 E) 12 Problem Kangur_2005_0304_19 (5 pts) http://www.mathkangaroo.org A picture frame was constructed using pieces of wood which all have the same width. What is the width of the frame if the inside perimeter of the frame is 8 decimeters less than its outside perimeter?
A) 1 dm B) 2 dm C) 4 dm D) 8 dm E) It depends on the size of the picture Problem Kangur_2005_0304_20 (5 pts) http://www.mathkangaroo.org In a trunk there are 5 chests, in each chest there are 3 boxes, and in each box there are 10 gold coins. The trunk, the chests, and the boxes are locked. At least how many locks need to be opened in order to take out 50 coins? A) 5 B) 6 C) 7 D) 8 E) 9 Problem Kangur_2005_0304_21 (5 pts) http://www.mathkangaroo.org The figure shows a rectangular garden with dimensions of 16 m by 20 m. The gardener has planted six identical flowerbeds (they are gray in the diagram). What is the perimeter of each of the flowerbeds?
A) 20 m B) 22 m C) 24 m D) 26 m E) 28 m Problem Kangur_2005_0304_22 (5 pts) http://www.mathkangaroo.org Mike chose a three-digit number and a two-digit number. The difference of these numbers is 989. What is their sum? A) 1001 B) 1010 C) 2005 D) 1000 E) 1009 Problem Kangur_2005_0304_23 (5 pts) http://www.mathkangaroo.org Five cards are laying on the table in the order: 5, 1, 4, 3, 2 as shown in the top row of the picture. They need to be placed in the order shown in the bottom row. In each move, any two cards may be switched. What is the least number of moves that need to be made?
A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2005_0304_24 (5 pts) http://www.mathkangaroo.org Which of the cubes has the plan shown in the picture below?
A)
B)
C)
D)
E)
Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2004_0304_1 (3 pts) http://www.mathkangaroo.org 2001+ 2002 + 2003 + 2004 + 2005 = A) 1,015 B) 5,010 C) 10,150 D) 11,005 E) 10,015 Problem Kangur_2004_0304_2 (3 pts) http://www.mathkangaroo.org Marek was 4 years old when his sister was born. Today he blew out all 9 candles on his birthday cake. What is the difference between Marek's and his sister's age today? A) 4 years B) 5 years C) 9 years D) 13 years E) 14 years Problem Kangur_2004_0304_3 (3 pts) http://www.mathkangaroo.org The picture below shows a road from town A to town B (indicated by solid line) and a detour (marked by a dash line) caused by renovation of the section CD. How many kilometres longer is the road from town A to town B because of the detour now?
A) 3 km B) 5 km C) 6 km D) 10 km E) This cannot be calculated. Problem Kangur_2004_0304_4 (3 pts) http://www.mathkangaroo.org
Which of the results below is not identical to the difference 671 - 389? A) 771 - 489 B) 681 - 399 C) 669 - 391 D) 1871 - 1589 E) 600 - 318 Problem Kangur_2004_0304_5 (3 pts) http://www.mathkangaroo.org There were some birds sitting on the telegraph wire. At one moment, 5 of them flied away and after some time, 3 birds came back. At that time there were 12 birds sitting on the wire. How many birds were there at the very beginning? A) 8 B) 9 C) 10 D) 12 E) 14 Problem Kangur_2004_0304_6 (3 pts) http://www.mathkangaroo.org Which numbers are inside a rectangle and inside a circle but not inside a triangle at the same time?
A) 5 and 11 B) 1 and 10 C) 13 D) 3 and 9 E) 6, 7 and 4 Problem Kangur_2004_0304_7 (3 pts) http://www.mathkangaroo.org Buildings on Color Street are numbered from 1 to 5 (see the picture). Each building is colored with one of the following colors: blue, red, yellow, pink, and green. It is known that: - The red building neighbours with the blue one only. - The blue building is between the red one and the green one.
What is the color of the building numbered with 3?
A) Blue B) Red C) Yellow D) Pink E) Green Problem Kangur_2004_0304_8 (3 pts) http://www.mathkangaroo.org How many white squares need to be shaded so that the number of shaded squares equals exactly to half of the number of white squares?
A) 2 B) 3 C) 4 D) 6 E) It is impossible to calculate it. Problem Kangur_2004_0304_9 (4 pts) http://www.mathkangaroo.org Five identical sheets of a plastic rectangles were divided into white and black squares. Which of the sheets from A) to E) has to be covered with the sheet to the right in order to get totally black rectangle?
A)
B)
C)
D)
Problem Kangur_2004_0304_10 (4 pts) http://www.mathkangaroo.org
E)
The scales in the pictures had been balanced. There are pencils and a pen on the arms of the scales. What is the weight of the pen in grams?
A) 6 g B) 7 g C) 8 g D) 9 g E) 10 g Problem Kangur_2004_0304_11 (4 pts) http://www.mathkangaroo.org I notice four clocks on the wall (see the picture). Only one of them shows correct time. One of them is 20 minutes ahead, another is 20 minutes late, and the other is stopped. What is the time at the moment?
A) 4:45 B) 5:05 C) 5:25 D) 5:40 E) 12:00 Problem Kangur_2004_0304_12 (4 pts) http://www.mathkangaroo.org Ella brought a basket of apples and oranges for a birthday party. Guests ate half of the apples and the third part of the oranges. In the basket remained: A) Half of all fruits B) More than half of all fruits C) Less than half of all fruits
D) A third part of all fruits E) Less than a third part of all fruits Problem Kangur_2004_0304_13 (4 pts) http://www.mathkangaroo.org Ania divided a certain number by 10 instead of multiplying it by 10. As a result she got 600. What would the result have been if she hadn't made that mistake? A) 6 B) 60 C) 600 D) 6,000 E) 60,000 Problem Kangur_2004_0304_14 (4 pts) http://www.mathkangaroo.org Kathy found a book, which was lack of certain number of sheets. When she opened the book she saw number 24 on the left side and number 45 on the right side. How many sheets between those sides were missing? A) 9 B) 10 C) 11 D) 20 E) 21 Problem Kangur_2004_0304_15 (4 pts) http://www.mathkangaroo.org Eva is 52 days older than her girlfriend Ania. Eva had her birthday on Tuesday in March of this year. On which day of the week will Ania celebrate her birthday this year? A) Monday B) Tuesday C) Wednesday D) Thursday E) Friday Problem Kangur_2004_0304_16 (4 pts) http://www.mathkangaroo.org Into the squares of diagram numbers were placed so that the sum of the numbers in the first row is equal to so that the sum of the numbers in the first row is equal to 3, the sum of the numbers in the second row is equal to 8, and the sum of the numbers in the first column is equal to 4. What is the sum of the numbers in the second column?
A) 4 B) 6 C) 7
D) 8 E) 11 Problem Kangur_2004_0304_17 (5 pts) http://www.mathkangaroo.org The cube (see the picture) is colored with three colors so that every side of this cube is one color and every two opposite sides are the same color. From which of the patterns below this kind of cube can be made?
Problem Kangur_2004_0304_18 (5 pts) http://www.mathkangaroo.org Four square tiles were arranged in a way shown in the picture. The lengths of the sides of two tiles are indicated in the picture. What is the length of the side of the largest tile?
A) 24 B) 56 C) 64 D) 81 E) 100 Problem Kangur_2004_0304_19 (5 pts) http://www.mathkangaroo.org Girls and boys from Maria's and Mathew's class have formed a line. There are 16 students on Maria's right, and Mathew is among them. There are 14 students on Mathew's left, and Maria is among them. There are 7 students between Maria and Mathew. How many students are in this class? A) 37 B) 30 C) 23 D) 22 E) 16 Problem Kangur_2004_0304_20 (5 pts) http://www.mathkangaroo.org The sum of the digits of the 10-digit number is 9.What is the product of the digits of this number?
A) 0 B) 1 C) 45 D) 9 x 8 x 7 x ... x 2 x 1 E) 10 Problem Kangur_2004_0304_21 (5 pts) http://www.mathkangaroo.org Out of 125 small, white and black cubes, the big cube was formed (see the picture). Every two adjacent cubes have different colors. The vertices of the big cube are black. How many white cubes does the big cube contain?
A) 62 B) 63 C) 64 D) 65 E) 68 Problem Kangur_2004_0304_22 (5 pts) http://www.mathkangaroo.org A lottery-ticket was 4 dollars. Three boys: Paul, Peter, and Robert made a contribiution and bought two tickets. Paul gave 1 dollar, Peter gave 3 dollars, and Robert gave 4 dollars. One of the tickets they bought was worth 1000 dollars. Boys shared the award fairly, meaning, proportionally to their contributions. How much did Peter receive? A) 300 B) 375 C) 250 D) 750 E) 425 Problem Kangur_2004_0304_23 (5 pts) http://www.mathkangaroo.org In three soccer games the Dziobak's team scored three goals and lost one. For every game won the team gets 3 points, for a tie it gets 1 point, and for the game lost it gets 0 points. For sure, the number of points the team earned in those three games was not equal to which of the following numbers? A) 7 B) 6 C) 5 D) 4 E) 3
Problem Kangur_2004_0304_24 (5 pts) http://www.mathkangaroo.org In every white section of a diagram, the products of two numbers from grey sections - one from above and one from the left - was placed (for example: 42 = 7 · 6 ). Some of these products are represented by letters. Which two letters represent the same number?
A) L and M B) T and N C) R and P D) K and P E) M and S Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2003_0304_1 (3 pts) http://www.mathkangaroo.org The picture below shows the letter U drawn on grid paper. How many squares does the letter U cover?
A) 10 B) 8 C) 11 D) 13 E) 12 Problem Kangur_2003_0304_2 (3 pts) http://www.mathkangaroo.org What is the sum of 0 + 1 + 2 + 3 + 4 - 3 - 2 - 1 - 0? A) 0 B) 2 C) 4
D) 10 E) 16 Problem Kangur_2003_0304_3 (3 pts) http://www.mathkangaroo.org The first train car, right behind the engine, contains 10 boxes. In each of the other cars there are twice as many boxes as in the car in front of it. How many boxes are there in the fifth car?
A) 100 B) 120 C) 140 D) 160 E) 180 Problem Kangur_2003_0304_4 (3 pts) http://www.mathkangaroo.org Zosia is drawing kangaroos. The first one is blue, the next one green, followed by red, and finally yellow, and then again blue, green, red, yellow, and so on, in the same order. What color will the seventeenth kangaroo be? A) Blue B) Green C) Red D) Black E) Yellow Problem Kangur_2003_0304_5 (3 pts) http://www.mathkangaroo.org In the teachers' lounge there are 6 tables with 4 chairs by each one, 4 tables with 2 chairs by each, and 3 tables with 6 chairs by each. How many chairs are there in the lounge? A) 40 B) 25 C) 50 D) 36 E) 44 Problem Kangur_2003_0304_6 (3 pts) http://www.mathkangaroo.org In one of these pictures, there are three times as many hearts as other shapes. Which picture is it?
A)
B)
C)
D)
E)
Problem Kangur_2003_0304_7 (3 pts) http://www.mathkangaroo.org A rectangle with dimensions 7 x 4 was outlined on grid paper. How many squares will a diagonal of this rectangle intersect?
A) 8 B) 9 C) 10 D) 11 E) 12 Problem Kangur_2003_0304_8 (3 pts) http://www.mathkangaroo.org The figure presented in the picture, made with identical cubes, weighs 189 grams. How much does one cube weigh?
A) 29 g B) 25 g C) 21 g D) 19 g E) 17 g Problem Kangur_2003_0304_9 (4 pts) http://www.mathkangaroo.org Peter wrote out consecutive natural numbers starting with 3 until he had written 35 digits. What was the greatest number that Peter wrote? A) 12 B) 22 C) 23 D) 28 E) 35 Problem Kangur_2003_0304_10 (4 pts) http://www.mathkangaroo.org Anna fell asleep at 9:30 PM and woke up at 6:45 AM the next day. Her little brother Peter slept 1 hour and 50 minutes longer. How long did Peter sleep?
A) 30 hr 5 min B) 11 hr 35 min C) 11 hr 5 min D) 9 hr 5 min E) 8 hr 35 min Problem Kangur_2003_0304_11 (4 pts) http://www.mathkangaroo.org A pattern, the beginning and the end of which is shown in the picture, is made up of alternating black and white bars. There are 17 bars altogether. The bars on both ends are black. How many white bars are there in the pattern?
A) 9 B) 16 C) 7 D) 8 E) 15 Problem Kangur_2003_0304_12 (4 pts) http://www.mathkangaroo.org Jumping Kangaroo is practicing for the animal Olympics. His longest jump during the training was 55 dm 50 mm long, but in the finals of the Olympics he won with a jump that was 123 cm longer. How long was Jumping Kangaroo's longest jump during the Olympics? A) 6 m 78 cm B) 5 m 73 cm C) 5 m 55 cm D) 11 m 28 cm E) 7 m 23 cm Problem Kangur_2003_0304_13 (4 pts) http://www.mathkangaroo.org Peter chose a certain number, then he subtracted 203 from it, then he added 2003 to that difference. His final result was 20003. What number did Peter choose at the beginning? A) 23 B) 17797 C) 18203 D) 21803 E) 22209 Problem Kangur_2003_0304_14 (4 pts) http://www.mathkangaroo.org Barbara likes to add the digits showing the current time on her electronic watch (for example, when the watch shows 21:17, she gets the number 11 as the result). What is the greatest sum she can get?
(Hint: in some countries and sometimes in USA, instead of telling it is 1P.M., people say it is 13:00. When it is 2P.M. they say it is 14:00, and when it is 12A.M., they say it is 24:00. In this problem 21:17 means 9:17P.M. Time expressed with this method is called military time sometimes.) A) 24 B) 36 C) 19 D) 25 E) 23 Problem Kangur_2003_0304_15 (4 pts) http://www.mathkangaroo.org Mark said to his friends, "If I had picked twice as many apples as I picked, I would have 24 more apples than I have now." How many apples did Mark pick? A) 48 B) 24 C) 42 D) 12 E) 36 Problem Kangur_2003_0304_16 (4 pts) http://www.mathkangaroo.org Points A, B, C, D all of which lie on a straight line, are marked in the figure below. The distance between points A and C is 10 m, between B and D is 15 m, and between A and D is 22 m. What is the distance between points B and C?
A) 1 m B) 2 m C) 3 m D) 4 m E) 5 m Problem Kangur_2003_0304_17 (5 pts) http://www.mathkangaroo.org There are 29 students in the class. 12 of the students have a sister and 18 of the students have a brother. In this class, only Tania, Barbara, and Anna do not have any siblings. How many students from this class have both a brother and a sister? A) None B) 1 C) 3 D) 4 E) 6 Problem Kangur_2003_0304_18 (5 pts) http://www.mathkangaroo.org
Peter has 11 pieces of paper. He cut some of them into three parts and now he has 29 pieces of paper. How many pieces of paper did he cut? A) 3 B) 2 C) 8 D) 11 E) 9 Problem Kangur_2003_0304_19 (5 pts) http://www.mathkangaroo.org Peter bought 3 kinds of cookies: large, medium, and small. The large cookies cost 4 zlotys each, the medium: 2 zlotys each, and the small: 1 zloty each. (A zloty is the Polish unit of money.) Altogether, Peter bought 10 cookies and paid 16 zlotys. How many large cookies did he buy? A) 5 B) 4 C) 3 D) 2 E) 1 Problem Kangur_2003_0304_20 (5 pts) http://www.mathkangaroo.org Christopher built a rectangular prism using red and blue cubes of identical size. The outer walls of this prism are red but all the inner cubes are blue. How many blue cubes did Christopher use in this construction?
A) 12 B) 24 C) 36 D) 40 E) 48 Problem Kangur_2003_0304_21 (5 pts) http://www.mathkangaroo.org Jurek is planning to buy some basketballs. If he were to buy 5 balls, he would have 10 zlotys left over, and if he were to buy 7 balls, he would have to borrow 22 zlotys. (A zloty is the Polish unit of money.) How much does one basketball cost?
A) 11 B) 16 C) 22 D) 26 E) 32 Problem Kangur_2003_0304_22 (5 pts) http://www.mathkangaroo.org Mark built a rectangular prism using 3 blocks, each of which is made up of 4 small cubes connected in various ways. Two of the blocks are shown in the picture. Which is the third, white block, of which only two sides are visible?
A)
B)
C)
D)
E)
Problem Kangur_2003_0304_23 (5 pts) http://www.mathkangaroo.org From a square puzzle, two pieces, which made up the shaded region, were cut out (see the figure). Which two of the pieces below are these?
A) 1 and 3 B) 2 and 4 C) 2 and 3 D) 1 and 4 E) 3 and 4 Problem Kangur_2003_0304_24 (5 pts) http://www.mathkangaroo.org
At the toy store, among other things, you can buy dogs, bears, and kangaroos. Three dogs and two bears together cost as much as four kangaroos. For the same amount of money you can buy one dog and three bears. Then: A) A dog is twice as expensive as a bear. B) A bear is twice as expensive as a dog. C) The prices of a dog and of a bear are identical. D) A bear is three times as expensive as a dog. E) A dog is three times as expensive as a bear. Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2002_0304_1 (3 pts) http://www.mathkangaroo.org Which of the squares below should be put into the picture to the right, to get the symbol of our competition?
A)
B)
C)
D)
E) Problem Kangur_2002_0304_2 (3 pts) http://www.mathkangaroo.org After we simplify 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2 - 2 + 2 what will be the result?
A) 0 B) 2 C) 4 D) 12 E) 20 Problem Kangur_2002_0304_3 (3 pts) http://www.mathkangaroo.org Andrzej received three cars, four balls, three teddy bears, ten pens, two chocolate bars, and a book for his birthday. How many items did he get in all? A) 15 B) 17 C) 20 D) 23 E) 27 Problem Kangur_2002_0304_4 (3 pts) http://www.mathkangaroo.org A square was divided into pieces (see the picture). Which of the following pieces does not occur in this divided square?
A)
B)
C)
D)
E) Problem Kangur_2002_0304_5 (3 pts) http://www.mathkangaroo.org
Julia, Kasia, Zuzanna, and Helena have their birthdays on March 1st, May 17th, July 20th, and March 20th. Kasia and Zuzanna were born in the same month. Julia and Zuzanna were born on the same day of a month. Which of the girls was born on May 17th? A) Julia B) Kasia C) Zuzanna E) Helena E) It cannot be determined from the given informfation. Problem Kangur_2002_0304_6 (3 pts) http://www.mathkangaroo.org A human heart beats an average of 70 times per minute. On average how many times does it beat during one hour? A) 42,000 B) 7,000 C) 4,200 D) 700 E) 420 Problem Kangur_2002_0304_7 (3 pts) http://www.mathkangaroo.org Quadrilateral ABCD is a square and its side is 10 cm long. Quadrilateral ATMD is a rectangle and its shorter side is 3 cm. What is the difference between the sum of the lengths of all the sides of the square and the sum of the lengths of all the sides of the rectangle?
A) 14 cm B) 10 cm C) 7 cm D) 6 cm E) 4 cm Problem Kangur_2002_0304_8 (3 pts) http://www.mathkangaroo.org Which of the figures below (see the picture) couldn't be made with folding a rectangular sheet just once?
A)
B)
C)
D)
E) Problem Kangur_2002_0304_9 (4 pts) http://www.mathkangaroo.org Houses on the street where John lives are numbered from 1 to 24. How many times does the digit 2 appear in the numbering of those houses? A) 2 B) 4 C) 8 D) 16 E) 32 Problem Kangur_2002_0304_10 (4 pts) http://www.mathkangaroo.org There are six identical oranges on one scale of the balance and two identical melons on the other scale. After we put one melon on the scale with the oranges, the scales will be balanced. How many oranges weigh as much as one melon?
A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2002_0304_11 (4 pts) http://www.mathkangaroo.org
This picture below is a sketch of a castle. Which of the lines below does not belong to the sketch?
A)
B) C) D) E) Problem Kangur_2002_0304_12 (4 pts) http://www.mathkangaroo.org We add 17 to the smallest two-digit number and then we divide the sum by the largest onedigit number. What is the result? A) 3 B) 6 C) 9 D) 11 E) 27 Problem Kangur_2002_0304_13 (4 pts) http://www.mathkangaroo.org In a certain ancient country the numbers: one, ten, and sixty were expressed with the following symbols: one, ten, sixty. Using those symbols people were writing down other numbers, for example the number 22 was written as Which of the following notations represents the number 124 ? A) B) C)
D) E) Problem Kangur_2002_0304_14 (4 pts) http://www.mathkangaroo.org A face of a clock was divided into four parts. The sums of the numbers in each of those parts are consecutive E numbers. Which of the following pictures satisfies this rule?
A)
B)
C)
D)
E) Problem Kangur_2002_0304_15 (4 pts) http://www.mathkangaroo.org Klara and Zosia had 60 matches altogether. Klara took as many matches as she needed to build a triangle, each side 6 matches long. Zosia used the remaining matches to build a rectangle, which had one side equal to 6 matches. How many matches long is each of the longer sides of this rectangle? A) 9 B) 12 C) 15 D) 18 E) 30 Problem Kangur_2002_0304_16 (4 pts) http://www.mathkangaroo.org
Three kangaroos: Miki, Niki, and Oki participated in a competition. Jumping at the same speed, they jumped along the lines you can see in the picture. Only one of the following sentences A, B, C, D and E is true. Which one?
A) Miki and Oki finished at the same time. B) Niki finished first. C) Oki finished last. D) All kangaroos finished at the same time. E) Miki and Niki finished at the same time. Problem Kangur_2002_0304_17 (5 pts) http://www.mathkangaroo.org Each boy: Mietek, Mirek, Pawel, and Zbyszek has exactly one of the following animals: a cat, a dog, a gold fish, and a canary-bird. Mirek has a pet with fur. Zbyszek has a pet with four legs. Pawel has a bird, and Mietek and Mirek don't like cats. Which of the following sentences is not true? A) Zbyszek has a dog. B) Pawel has a canary. C) Mietek has a golden fish. D) Zbyszek has a cat. E) Mirek has a dog. Problem Kangur_2002_0304_18 (5 pts) http://www.mathkangaroo.org Marysia leaves her house at 6:55 and arrives at school at 7:32. Zosia needs 12 minutes less than Marysia to get to school. Yesterday Zosia showed up at school at 7:45. What time did she leave her house? A) At 7:07 B) At 7:20 C) At 7:25 D) At 7:30 E) At 7:33 Problem Kangur_2002_0304_19 (5 pts) http://www.mathkangaroo.org Robert had a certain number of identical cubes. He glued a tunnel using half of his blocks (see Picture 1). With some of the remaining cubes he formed a pyramid (see Picture 2). How
many blocks were not used to build those structures?
A) 34 B) 28 C) 22 D) 18 E) 15 Problem Kangur_2002_0304_20 (5 pts) http://www.mathkangaroo.org Daughter is 3 years old, and her mother is 28 years older than the daughter. How many years later will the mother be three times older than her daughter? A) 9 B) 12 C) 10 D) 1 E) 11 Problem Kangur_2002_0304_21 (5 pts) http://www.mathkangaroo.org A conductor wanted to make a trio consisting of a fiddler, a pianist, and a drummer. He had to choose one of two fiddlers, one of two pianists, and one of two drummers. He decided to try each of the possible trios. How many attempts did he have to make? A) 3 B) 4 C) 8 D) 24 E) 25 Problem Kangur_2002_0304_22 (5 pts) http://www.mathkangaroo.org One medal can be cut out from a golden square plate. If four medals are made from four plates, the remaining parts of those four plates can be used to make one more plate. What is the largest number of medals that could be formed when 16 plates are used? A) 17 B) 19 C) 20 D) 21 E) 32
Problem Kangur_2002_0304_23 (5 pts) http://www.mathkangaroo.org Twenty eight students from the fourth grade competed in the math competition. Each student earned a different number of points. The number of students who received more points than Tomek is two times smaller than the number of students who had less points than Tomek. In which position did Tomek finish that competition? A) 6th B) 7th C) 8th D) 9th E) 10th Problem Kangur_2002_0304_24 (5 pts) http://www.mathkangaroo.org An odometer in a car shows the number 187569 of passed kilometers. This number consists of all different digits. After passing how many kilometers will the odometer show a number consisting of all different digits again? A) After 777 km B) After 12,431 km C) After 431 km D) After 21 km E) After 11 km Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Grade 05-06 Problem Kangur_2005_0506_1 (3 pts) http://www.mathkangaroo.org A butterfly sat down on a correctly solved problem. What number did it cover up?
2005 + 205 = 3500 A) 1295 B) 1190 C) 1390 D) 1195 E) 1290 Problem Kangur_2005_0506_2 (3 pts) http://www.mathkangaroo.org Together, Anna and Olla have ten pieces of candy. Olla has two more pieces of candy than Anna. How many pieces of candy does Olla have? A) 8 B) 7 C) 6 D) 5 E) 4 Problem Kangur_2005_0506_3 (3 pts) http://www.mathkangaroo.org There are eight kangaroos in the diagram (see the picture). What is the least number of kangaroos that have to be moved to the empty boxes in order to have two kangaroos in each row and each column?
A) 0 B) 1 C) 2 D) 3 E) 4 Problem Kangur_2005_0506_4 (3 pts) http://www.mathkangaroo.org
Eva lives with her parents, a brother, a dog, two cats, two parrots, and four gold fish. How many legs do they have altogether? A) 40 B) 32 C) 28 D) 24 E) 22 Problem Kangur_2005_0506_5 (3 pts) http://www.mathkangaroo.org 2005 x 100 + 2005 = A) 2005002005 B) 20052005 C) 20072005 D) 202505 E) 22055 Problem Kangur_2005_0506_6 (3 pts) http://www.mathkangaroo.org An ant is walking from point A to point B on a cube along the indicated path. The edge of the cube is 12 cm long. How far does the ant need to travel?
A) 40 cm B) 48 cm C) 50 cm D) 60 cm E) 36 cm Problem Kangur_2005_0506_7 (3 pts) http://www.mathkangaroo.org
On a shelf, there are 24 balls in three colors: white, red and brown. of the rest of the balls are red. How many of them are brown? A) 4 B) 5 C) 6
of them are white, and
D) 7 E) 8 Problem Kangur_2005_0506_8 (3 pts) http://www.mathkangaroo.org There are five cards on the table, labeled with numbers 1 to 5 as shown in the top row. One move consists of switching two cards. How many moves do you need to make so that the cards are arranged in the way shown in the bottom row?
A) 2 B) 4 C) 1 D) 3 E) 5 Problem Kangur_2005_0506_9 (3 pts) http://www.mathkangaroo.org Tom picked a natural number and multiplied it by 3. Which number CANNOT be the result of this multiplication? A) 987 B) 444 C) 204 D) 105 E) 103 Problem Kangur_2005_0506_10 (3 pts) http://www.mathkangaroo.org How many hours is half of a third part of a quarter of 24 hours?
A) B) C) 1 D) 2 E) 3 Problem Kangur_2005_0506_11 (4 pts) http://www.mathkangaroo.org Eva cut a paper napkin into 10 pieces. She then also cut one of the pieces into 10 pieces. She repeated this process two more times. Into how many pieces did she cut the napkin?
A) 27 B) 30 C) 37 D) 40 E) 47 Problem Kangur_2005_0506_12 (4 pts) http://www.mathkangaroo.org Mowgli usually walks from home to the beach, and returns on an elephant. It takes him 40 minutes altogether. One day he traveled on the elephant from home to the beach and back, which took him 32 minutes. How much time would he need to travel the same distance on foot? A) 24 min B) 42 min C) 46 min D) 48 min E) 50 min Problem Kangur_2005_0506_13 (4 pts) http://www.mathkangaroo.org A rectangular garden with an area of 30 m² was divided into three rectangular sections of flowers, vegetables, and strawberries (some of the dimensions are shown in the diagram). What is the area of the vegetable section, if the flower part has an area of 10 m²?
A) 4 m² B) 6 m² C) 8 m² D) 10 m² E) 12 m² Problem Kangur_2005_0506_14 (4 pts) http://www.mathkangaroo.org Grandpa suggested dividing all peanuts between the family members in the following way: one person would get 5 kilos, two people would get 4 kilos each, four people would get 2 kilos each, two people would get 1.5 kilo each, and one person would not get any nuts. Grandma suggested dividing the peanuts equally among all of the family members. For how many people would the division suggested by Grandma be better than the one suggested by Grandpa?
A) 3 B) 4 C) 5 D) 6 E) 7 Problem Kangur_2005_0506_15 (4 pts) http://www.mathkangaroo.org How many two digit numbers are there, which can be expressed only by using different odd digits? A) 15 B) 20 C) 25 D) 30 E) 50 Problem Kangur_2005_0506_16 (4 pts) http://www.mathkangaroo.org Which of the cubes below represents the plan of the cube shown to the right?
Problem Kangur_2005_0506_17 (4 pts) http://www.mathkangaroo.org Sum of five consecutive natural numbers is equal to 2005. What is the greatest number among them? A) 401 B) 403 C) 404 D) 405 E) 2001 Problem Kangur_2005_0506_18 (4 pts) http://www.mathkangaroo.org What is the number of all divisors of number 100?
A) 3 B) 6 C) 7 D) 8 E) 9 Problem Kangur_2005_0506_19 (4 pts) http://www.mathkangaroo.org The frame of a rectangular painting was made out of wooden pieces of the same width. What is the width of those pieces if the outer perimeter of the frame is 8 decimeters longer than the inner perimeter?
A) 4dm B) 2dm C) 1dm D) 8dm E) The width depends on the dimensions of the painting. Problem Kangur_2005_0506_20 (4 pts) http://www.mathkangaroo.org How many more triangles than squares are shown in the picture?
A) 4 more B) 2 more C) 1 more D) 5 more E) 3 more Problem Kangur_2005_0506_21 (5 pts) http://www.mathkangaroo.org There are five containers in a treasure chest, in each container there are three boxes and in each box there are 10 golden coins. The treasure chest, the containers, and the boxes are all locked. How many locks do you need to open to get 50 coins?
A) 5 B) 7 C) 9 D) 6 E) 8 Problem Kangur_2005_0506_22 (5 pts) http://www.mathkangaroo.org What number should replace x, if we know that the number in the circle in the upper row is the sum of the numbers from the two circles right below it.
A) 32 B) 50 C) 55 D) 82 E) 100 Problem Kangur_2005_0506_23 (5 pts) http://www.mathkangaroo.org In a two-digit number, a is the tens digit and b is the ones digit. Which of the conditions below ensures that the number will be divisible by 6? A) a + b = 6 B) b = 6 a C) b = 5 a D) b = 2 a E) a = 2 b Problem Kangur_2005_0506_24 (5 pts) http://www.mathkangaroo.org A wooden cube with the length of its side equal to 3 dm was painted with 0.25 kg of paint. The cube was then cut up into unit cubes (side length of 1 dm). How much paint is needed to paint the unpainted sides of the little cubes? A) 1.25 kg B) 1 kg C) 0.75 kg D) 0.5 kg E) 0.25 kg Problem Kangur_2005_0506_25 (5 pts) http://www.mathkangaroo.org
Five circles have radii of the same length (see the picture). Four of them are touching the fifth circle, and their centers are the vertices of a square. The ratio of the area of the shaded region of the circles to the area of unshaded regions of the circles is:
A) 1 : 3 B) 1 : 4 C) 2 : 5 D) 2 : 3 E) 5 : 4 Problem Kangur_2005_0506_26 (5 pts) http://www.mathkangaroo.org From noon until midnight, Wise Cat sleeps under a chestnut tree. From midnight until noon he is awake telling stories. There is a note on that tree which says: "Two hours ago, Wise Cat was doing the same thing that he will be doing in an hour". How many hours, out of 24 hours, is the note true? A) 6 B) 12 C) 18 D) 3 E) 21 Problem Kangur_2005_0506_27 (5 pts) http://www.mathkangaroo.org Mark has 42 cubes with side length of 1 cm. He used them to construct a prism, the base of which has a perimeter of 18 cm. What is the height of that prism? A) 6 cm B) 5 cm C) 4 cm D) 3 cm E) 2 cm Problem Kangur_2005_0506_28 (5 pts) http://www.mathkangaroo.org On the board Peter wrote all the three-digit numbers that have the following properties: the digits in each of the numbers are different, the first digit is the square of the quotient of the second digit and the third digit. How many numbers did Peter write? A) 1 B) 2
C) 3 D) 4 E) 8 Problem Kangur_2005_0506_29 (5 pts) http://www.mathkangaroo.org Equilateral triangle ABC (all sides congruent) has an area equal to 1. A bigger triangle was constructed out of 49 of these triangles (see the picture). The area of the shaded region is equal to:
A) 20 B) 22.5 C) 23.5 D) 25 E) 32 Problem Kangur_2005_0506_30 (5 pts) http://www.mathkangaroo.org Mary, Dorothy, Sylvia, Ella, and Kathy are sitting on a bench in the park. Mary is not sitting on the farthest right side; Dorothy is not sitting the farthest to the left. Sylvia is not sitting the farthest to the left nor the farthest to the right. Kathy is not sitting next to Sylvia, and Sylvia is not sitting next to Dorothy. Ella is sitting to the right of Dorothy, but not necessarily next to her. Which girl is sitting the farthest to the right? A) It cannot be determined. B) Dorothy C) Sylvia D) Ella E) Kathy Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2004_0506_1 (3 pts) http://www.mathkangaroo.org How much is 1000 - 100 + 10 - 1?
A) 111 B) 900 C) 909 D) 990 E) 999 Problem Kangur_2004_0506_2 (3 pts) http://www.mathkangaroo.org In each of the little squares Karolina places one of the digits: 1, 2, 3, 4. She makes sure that in each row and each column each of these numbers is placed. In the figure below, you can see the way of filling these squares. What number should she put in the square marked with an x? 1
x 2
4 1 3 2 A) 1 B) 2 C) 3 D) 4 E) Cannot be determined. Problem Kangur_2004_0506_3 (3 pts) http://www.mathkangaroo.org (10 · 100) · (20 · 80) = A) 20,000 · 80,000 B) 2000 · 8000 C) 2000 · 80,000 D) 20,000 · 8000 E) 2000 · 800 Problem Kangur_2004_0506_4 (3 pts) http://www.mathkangaroo.org How many hours is 360,000 seconds? A) 3 hours B) 6 hours C) 8.5 hours D) 10 hours E) More than 90 hours. Problem Kangur_2004_0506_5 (3 pts) http://www.mathkangaroo.org What is the remainder when you divide 20042003 by 2004?
A) 0 B) 1 C) 2 D) 3 E) 2003 Problem Kangur_2004_0506_6 (3 pts) http://www.mathkangaroo.org Five identical sheets of a plastic rectangles were divided into white and black squares. Which of the sheets from A to E has to be covered with the sheet to the right in order to get totally black rectangle?
A)
B)
C)
D)
E)
Problem Kangur_2004_0506_7 (3 pts) http://www.mathkangaroo.org Which of the following numbers is not a factor of 2004? A) 3 B) 4 C) 6 D) 8 E) 12 Problem Kangur_2004_0506_8 (3 pts) http://www.mathkangaroo.org The three members of a rabbit family ate 73 carrots altogether during a week. The father ate five carrots more than the mother. Their son ate 12 carrots. How many carrots did mother eat in that week? A) 27 B) 28 C) 31 D) 33 E) 56 Problem Kangur_2004_0506_9 (3 pts) http://www.mathkangaroo.org
Nine bus stops are equally spaced along a bus route. The distance between the first stop and the third one is 600 m. How long is the bus route? A) 1800 m B) 2100 m C) 2400 m D) 2700 m E) 3000 m Problem Kangur_2004_0506_10 (3 pts) http://www.mathkangaroo.org What is the value of the expression 1 - (2 - (3 - (4 - 5) ) ) equal to? A) 0 B) -3 C) -9 D) 3 E) 9 Problem Kangur_2004_0506_11 (4 pts) http://www.mathkangaroo.org You are given two identical puzzle pieces and you are not allowed to turn them over. Which figure cannot be made out of these two pieces?
Problem Kangur_2004_0506_12 (4 pts) http://www.mathkangaroo.org Karol folds a sheet of paper in a half and then repeats this four more times. Then he makes a hole in the folded paper. How many holes does the sheet of paper have after unfolding?
A) 6 B) 10 C) 16 D) 20 E) 32 Problem Kangur_2004_0506_13 (4 pts) http://www.mathkangaroo.org
The different figures represent different digits. Find the digit corresponding to the square.
A) 9 B) 8 C) 7 D) 6 E) 5 Problem Kangur_2004_0506_14 (4 pts) http://www.mathkangaroo.org The weight of 3 apples and 2 oranges is 255 g. The weight of 2 apples and 3 oranges is 285 g. Each apple weighs the same and each orange weighs the same. What is the combined weight of 1 apple and 1 orange? A) 110 g B) 108 g C) 105 g D) 104 g E) 102 g Problem Kangur_2004_0506_15 (4 pts) http://www.mathkangaroo.org Tomek, Romek, Andrzej, and Michal said the following about a certain number: Tomek: "This number is equal to 9"; Romek: "This number is prime."; Andrzej: "This number is even."; Michal: "This number is equal to 15." Only one statement given either by Romek or Tomek is true, as well as only one statement given by either Andrzej or Michal is true. What number is it? A) 1 B) 2 C) 3 D) 9 E) 15 Problem Kangur_2004_0506_16 (4 pts) http://www.mathkangaroo.org What is the smallest number of the little squares that have to be shaded in order to get at least one axis of symmetry of the figure below?
A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2004_0506_17 (4 pts) http://www.mathkangaroo.org One corner of a cube was cut off. Which of the figure below represents the pattern of the cube after unfolding it?
Problem Kangur_2004_0506_18 (4 pts) http://www.mathkangaroo.org Four snails: Fin, Pin, Rin, and Tin are moving along identical rectangular tiles. The shape and length of each snail's trip is shown below. How many decimeters has snail Tin gone? Snail Fin has gone 25 dm. Snail Pin has gone 37 dm. Snail Rin has gone 38 dm. Snail Tin has gone ? dm
A) 27 dm B) 30 dm C) 35 dm D) 36 dm E) 40 dm Problem Kangur_2004_0506_19 (4 pts) http://www.mathkangaroo.org The Island of Turtles has an unusual weather system: Mondays and Wednesdays are rainy, Saturdays are foggy and the other days are sunny. A group of tourists would like to go on a 44-day long vacation to the island. Which day of the week should be the first day of their vacation in order to enjoy the most of the sunny days?
A) Monday B) Wednesday C) Thursday D) Friday E) Tuesday Problem Kangur_2004_0506_20 (4 pts) http://www.mathkangaroo.org The sum of two natural numbers is equal to 77. If the first number is multiplied by 8 and the second by 6, then those products are equal. What is the larger of these numbers? A) 23 B) 33 C) 43 D) 44 E) 54 Problem Kangur_2004_0506_21 (5 pts) http://www.mathkangaroo.org What is the number of all divisors of 2 · 3 · 5 · 7 ? A) 4 B) 14 C) 16 D) 17 E) 210 Problem Kangur_2004_0506_22 (5 pts) http://www.mathkangaroo.org
Ella and Ola had 70 mushrooms altogether. of Ella's mushrooms are brown and Ola's mushrooms are white. How many mushrooms did Ella have?
of
A) 27 B) 36 C) 45 D) 54 E) 10 Problem Kangur_2004_0506_23 (5 pts) http://www.mathkangaroo.org There are 11 fields in the picture. Number 7 is written in the first field and number 6 in the ninth field. What number has to be placed in the second field so that the sum of the numbers from every three consecutive fields is equal to 21?
A) 7 B) 8 C) 6
D) 10 E) 21 Problem Kangur_2004_0506_24 (5 pts) http://www.mathkangaroo.org The square below was divided into small squares. What part of the area of the shaded figure is the area of the figure that is not shaded?
A) B) C) D) E) Problem Kangur_2004_0506_25 (5 pts) http://www.mathkangaroo.org In a CD store two CDs have the same price. The price of the first CD was reduced by 5% and the price of the other one was increased by 15%. After this change the prices of the two CDs differed by $6.00. How much is the cheaper CD now? A) $1.50 B) $6.00 C) $28.50 D) $30.00 E) $34.50 Problem Kangur_2004_0506_26 (5 pts) http://www.mathkangaroo.org In the little squares of a big square the consecutive natural numbers are placed in a way shown in the figure. Which of the numbers below cannot be placed in the square with letter x?
A) 128 B) 256 C) 81 D) 121 E) 400 Problem Kangur_2004_0506_27 (5 pts) http://www.mathkangaroo.org
Ania divided number
by 3. What is the number of zeros in the quotient?
A) 670 B) 669 C) 668 D) 667 E) 665 Problem Kangur_2004_0506_28 (5 pts) http://www.mathkangaroo.org Imagine that you have 108 red balls and 180 green balls. The balls have to be packed in boxes in such a way that every box contains the same number of balls and there are balls of only one color in every box. What is the smallest number of boxes that you need? A) 288 B) 36 C) 18 D) 8 E) 1 Problem Kangur_2004_0506_29 (5 pts) http://www.mathkangaroo.org During a competition in the Kangaroo Summer Camp in Zakopane students were given 10 problems to solve. For each correct answer a student was given 5 points and for each incorrect one the student was loosing 3 points. Everybody solved all the problems. Mathew got 34 points, Philip got 10 points and John got 2 points. How many problems did they answer correctly all together? A) 17 B) 18 C) 15 D) 13 E) 21 Problem Kangur_2004_0506_30 (5 pts) http://www.mathkangaroo.org A right triangle with legs of length 6 cm and 8 cm was cut out of a paper and then folded along a straight line. Which of the numbers below can express the area of the resulting polygon?
A) 9 cm² B) 12 cm² C) 18 cm² D) 24 cm² E) 30 cm² Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2003_0506_1 (3 pts) http://www.mathkangaroo.org Which of the following numbers is greatest? A) 2 + 0 + 0 + 3 B) 2 x 0 x 0 x 3 C) ( 2 + 0 ) x ( 0 + 3 ) D) 20 x 0 x 3 E) ( 2 x 0 ) + ( 0 x 3 ) Problem Kangur_2003_0506_2 (3 pts) http://www.mathkangaroo.org Zosia is drawing flowers of different colors. The first flower is blue, then white, red, yellow, and again blue, white, red, yellow, and so on in the same order. What is the color of the twenty ninth flower drawn by Zosia? A) Blue B) White C) Red D) Pink E) Yellow Problem Kangur_2003_0506_3 (3 pts) http://www.mathkangaroo.org How many integers are there on the number line between the numbers 2.09 and 15.3? A) 13 B) 14 C) 11 D) 12 E) Infinitely many Problem Kangur_2003_0506_4 (3 pts) http://www.mathkangaroo.org The least positive integer which, is divisible by 2, 3, and 4, is: A) 1 B) 6 C) 12
D) 24 E) 36 Problem Kangur_2003_0506_5 (3 pts) http://www.mathkangaroo.org Two of the numbers located on the two circles (see the picture) are represented by letters A and B. The sum of the numbers on each circle is equal to 55. What number is represented by letter A?
A) 9 B) 10 C) 13 D) 16 E) 17 Problem Kangur_2003_0506_6 (3 pts) http://www.mathkangaroo.org Tomek has 9 bills worth 100 zlotys each, 9 bills worth 10 zlotys each, and 10 coins worth 1 zloty each. How much money does Tomek have? (a zloty [zl] is a monetary unit in Poland) A) 1,000 zl B) 991 zl C) 9, 910 zl D) 9,901 zl E) 99, 010 zl Problem Kangur_2003_0506_7 (3 pts) http://www.mathkangaroo.org A square with the length of side equal to x consists of a square with an area of 81 cm², two rectangles with areas of 18 cm² each, and a small square. What is the value of x?
A) 2 cm B) 7 cm C) 9 cm
D) 10 cm E) 11 cm Problem Kangur_2003_0506_8 (3 pts) http://www.mathkangaroo.org
The value of the expression
is equal to:
A) 2003 B) C) 3 D) E) 6009 Problem Kangur_2003_0506_9 (3 pts) http://www.mathkangaroo.org Basia likes to add the digits that indicate the actual time on her electronic watch (for example, when the watch shows 21:17, she gets the sum equal to 11). What is the greatest sum she can get? (Hint: in some countries and sometimes in USA, instead of telling it is 1P.M., people say it is 13:00. When it is 2P.M. they say it is 14:00, and when it is 12A.M., they say it is 24:00. In this problem 21:17 means 9:17P.M. Time expressed with this method is called military time sometimes.) A) 24 B) 36 C) 19 D) 25 E) 28 Problem Kangur_2003_0506_10 (3 pts) http://www.mathkangaroo.org The picture shows Clown Jan dancing on two balls and a cube. The radius of the lower ball is 6 dm, and the radius of the upper ball is three times shorter. The edge of the cube is 4 dm longer than the radius of the upper ball. At what height is Jan dancing?
A) 14 dm B) 20 dm C) 22 dm
D) 24 dm E) 28 dm Problem Kangur_2003_0506_11 (4 pts) http://www.mathkangaroo.org Let AC = 10 m, BD = 15 m, AD = 22 m (see the figure below). What is length of segment BC is equal to?
A) 1 m B) 2 m C) 3 m D) 4 m E) 5 m Problem Kangur_2003_0506_12 (4 pts) http://www.mathkangaroo.org How many shortest distances along the edges of the cube are there that connect vertex A with the opposite vertex B?
A) 4 B) 6 C) 3 D) 12 E) 16 Problem Kangur_2003_0506_13 (4 pts) http://www.mathkangaroo.org From a square puzzle two pieces are cut out. These two pieces made the shaded region, (see the figure). Among the four figures below, which are these two pieces?
A) 1 and 4 B) 2 and 4 C) 2 and 3 D) 1 and 3 E) 3 and 4 Problem Kangur_2003_0506_14 (4 pts) http://www.mathkangaroo.org We add two different numbers chosen from the numbers: 1, 2, 3, 4, 5. How many different sums can we get? A) 5 B) 6 C) 7 D) 8 E) 9 Problem Kangur_2003_0506_15 (4 pts) http://www.mathkangaroo.org The figure in the picture consists of 7 squares. Square A has the greatest area, and square B the smallest area. The lengths of two of the squares are given. How many B squares will it take to fill up square A completely?
A) 16 B) 25 C) 36 D) 49 E) It is impossible. Problem Kangur_2003_0506_16 (4 pts) http://www.mathkangaroo.org A certain bar code consists of 17 black bars. A white bar divides each two black bars. The first bar and the last bar in the code are black. There are two kinds of black bars: wide and narrow. The number of white bars is 3 more than the number of wide black bars. How many narrow black bars are there in this bar code?
A) 1 B) 2 C) 3
D) 4 E) 5 Problem Kangur_2003_0506_17 (4 pts) http://www.mathkangaroo.org Ewa has 20 balls of four colors: yellow, green, blue, and black. 17 of them are not green, 5 are black, and 12 are not yellow. How many blue balls does Ewa have? A) 3 B) 4 C) 6 D) 7 E) 8 Problem Kangur_2003_0506_18 (4 pts) http://www.mathkangaroo.org There are 17 trees on one side of the street on Tomek's way from his house to school. One day Tomek marked these trees with white chalk in the following way: on the way from his house to the school he marked every other tree, starting with the first one. On his way back home he marked every third tree, starting with the first one. How many trees were not marked? A) 4 B) 5 C) 6 D) 7 E) 8 Problem Kangur_2003_0506_19 (4 pts) http://www.mathkangaroo.org Today the date is 3.20.2003 and the time is 20:03 (8:03 P.M.) What will be the date after 2003 minutes? A) 3.21.2003 B) 3.22.2003 C) 3.23.2003 D) 4.21.2003 E) 4.22.2003 Problem Kangur_2003_0506_20 (4 pts) http://www.mathkangaroo.org What is the digit of ones in the number 20032003? A) 7 B) 1 C) 9 D) 5 E) 3 Problem Kangur_2003_0506_21 (5 pts) http://www.mathkangaroo.org
With how many zeros does the product of the consecutive natural numbers from 1 to 50 end? A) 5 B) 10 C) 12 D) 20 E) 50 Problem Kangur_2003_0506_22 (5 pts) http://www.mathkangaroo.org The square ABCD consists of a white square and four shaded rectangles. Each of the rectangles has a perimeter of 40 cm. What is the area of square ABCD?
A) 100 cm² B) 200 cm² C) 160 cm² D) 400 cm² E) 80 cm² Problem Kangur_2003_0506_23 (5 pts) http://www.mathkangaroo.org We have six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we select three of these segments to build a triangle? A) 1 B) 3 C) 5 D) 6 E) 10 Problem Kangur_2003_0506_24 (5 pts) http://www.mathkangaroo.org Piotrek is writing the numbers from 0 to 109 into a five-column table using a rule which is easy to understand (see the picture below). Which of the pieces below can not be filled in
with numbers to fit Piotrek's table?
Problem Kangur_2003_0506_25 (5 pts) http://www.mathkangaroo.org In the figure, the beginning part of the path from point A to point B is shown. How long is the whole path?
A) 10,200 cm B) 2,500 cm C) 909 cm D) 10,100 cm E) 9,900 cm Problem Kangur_2003_0506_26 (5 pts) http://www.mathkangaroo.org At 3:00 o'clock the minute hand and the hour hand make a right angle. What will the measure of the angle between these hands be after 10 minutes? A) 90° B) 30° C) 80° D) 60° E) 35° Problem Kangur_2003_0506_27 (5 pts) http://www.mathkangaroo.org
In the addition, every square stands for a certain digit, every triangle stands for another specific digit, and every circle denotes yet another digit. What is the sum of the numbers represented by the square and the circle?
A) 6 B) 7 C) 8 D) 9 E) 13 Problem Kangur_2003_0506_28 (5 pts) http://www.mathkangaroo.org The shaded figure at the picture consists of five identical isosceles right triangles (see the figure below). What is the area of the shaded figure?
A) 20 cm² B) 25 cm² C) 35 cm² D) 45 cm² E) 60 cm² Problem Kangur_2003_0506_29 (5 pts) http://www.mathkangaroo.org Red and green dragons lived in a cave. Every red dragon had 6 heads, 8 legs, and 2 tails. Every green dragon had 8 heads, 6 legs, and 4 tails. There were 44 tails altogether, and there were 6 less green legs than red heads. How many red dragons lived in the cave? A) 6 B) 7 C) 8 D) 9 E) 10 Problem Kangur_2003_0506_30 (5 pts) http://www.mathkangaroo.org Ania has 9 crayons in a box. At least one of them is blue. From every 4 crayons at least two are of the same color, and from every 5 crayons at most three are of the same color. How many blue crayons are in this box?
A) 2 B) 3 C) 4 D) 1 E) 5 Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2002_0506_1 (3 pts) http://www.mathkangaroo.org The number 2002 read from left to right and from right to left is the same. Which number from the numbers below does not have this property? A) 1991 B) 2323 C) 2112 D) 2222 E) 4334 Problem Kangur_2002_0506_2 (3 pts) http://www.mathkangaroo.org The picture below is a sketch of a castle. Which of the lines below does not belong to the sketch?
A)
B) C) D) E) Problem Kangur_2002_0506_3 (3 pts) http://www.mathkangaroo.org Mr. and Mrs. Kowalski have three daughters. Each of them has two brothers. How many children does the Kowalski family have?
A) 9 B) 7 C) 6 D) 5 D) 11 Problem Kangur_2002_0506_4 (3 pts) http://www.mathkangaroo.org In which number below is the square of the tens digit equal to the triple of the sum of the digits of hundreds and ones? A) 192 B) 741 C) 385 D) 138 E) 231 Problem Kangur_2002_0506_5 (3 pts) http://www.mathkangaroo.org What is the product 22 · 22000 · 2 equal to? (· denotes multiplication) A) 24000 B) 22002 C) 22003 D) 24002 E) 24001 Problem Kangur_2002_0506_6 (3 pts) http://www.mathkangaroo.org On which string is the number of black hearts equal to two thirds of the number of all the hearts on that string?
A) B) C) D) E) Problem Kangur_2002_0506_7 (3 pts) http://www.mathkangaroo.org Which of the numbers below is the greatest? (· denotes multiplication, : denotes division)
A) 10 · 0.001 · 100 B) 0.01 : 100 C) 100 : 0.01 D) 10,000 · 100 : 10 E) 0.1 · 0.01 · 10,000 Problem Kangur_2002_0506_8 (3 pts) http://www.mathkangaroo.org What is the area of the figure in the picture below?
A) 43 B) 88 C) 58 D) 30 E) 15 Problem Kangur_2002_0506_9 (3 pts) http://www.mathkangaroo.org The area of a certain rectangle is equal to 1 m². What is the area of a triangle that was cut off from that rectangle along the line connecting the midpoints of the two adjacent sides? A) 33 dm² B) 25 dm² C) 40 dm² D) 3,750 cm² E) 1,250 cm² Problem Kangur_2002_0506_10 (3 pts) http://www.mathkangaroo.org We subtracted the smallest three-digit number with all different digits from the greatest threedigit number with all different digits. What is the result? A) 864 B) 885 C) 800 D) 899 E) Other number Problem Kangur_2002_0506_11 (4 pts) http://www.mathkangaroo.org
Figures I, II, III and IV are squares. The perimeter of square I is equal to 16 m, and the perimeter of square II is equal to 24 m. What is the perimeter of square IV ?
A) 56 m B) 60 m C) 64 m D) 72 m E) 80 m Problem Kangur_2002_0506_12 (4 pts) http://www.mathkangaroo.org One medal can be cut out from a golden square plate. If four medals are made from four plates, the remaining parts of those four plates can be used to make one more plate. What is the largest number of medals that could be formed when 64 plates are used? A) 85 B) 64 C) 80 D) 84 E) 100 Problem Kangur_2002_0506_13 (4 pts) http://www.mathkangaroo.org Rectangle ABCD (see the picture) is built out of 24 little squares with the length of each side equal to 1. What is the area of triangle ALM?
A) 5 B) 6 C) 7 D) 8 E) Other Problem Kangur_2002_0506_14 (4 pts) http://www.mathkangaroo.org
In the picture below the coordinates of points A and B were indicated. What are the coordinates of points C and D if AB = 2 BC, BC = 2 CD ?
A) 24 and 32 B) 24 and 28 C) 24 and 26 D) 22 and 24 E) 22 and 23 Problem Kangur_2002_0506_15 (4 pts) http://www.mathkangaroo.org Mirek has 9 sticks with the lengths of 1 dm, 2 dm, 3 dm, 4 dm, 5 dm, 6 dm, 7 dm, 8 dm, 9 dm. With the sticks he builds triangles of which each side is built with one stick. How many triangles with a side of 1 dm can be built with those sticks? A) 6 B) 3 C) 2 D) 1 E) 0 Problem Kangur_2002_0506_16 (4 pts) http://www.mathkangaroo.org How many convex angles with different measures are made by the rays with P as the starting point (see the picture)?
A) 4 B) 6 C) 8 D) 10 E) 11 Problem Kangur_2002_0506_17 (4 pts) http://www.mathkangaroo.org How many different three-digit numbers divisible by 25 can be made with the digits 0, 3, 5, 7 if the digits can be repeated? A) 16 B) 9 C) 81
D) 64 E) 3 Problem Kangur_2002_0506_18 (4 pts) http://www.mathkangaroo.org Each boy: Mietek, Mirek, Pawel, and Zbyszek has exactly one of the following animals: a cat, a dog, a gold fish, and a canary-bird. Mirek has a pet with fur. Zbyszek has a pet with four legs. Pawel has a bird, and Mietek and Mirek don't like cats. Which of the following sentences is not true? A) Zbyszek has a dog. B) Pawel has a canary. C) Mietek has a golden fish. D) Zbyszek has a cat. E) Mirek has a dog. Problem Kangur_2002_0506_19 (4 pts) http://www.mathkangaroo.org The next day after his birthday Jas said: "The day after tomorrow will be Thursday." On what day of the week did Jas have his birthday? A) On Monday B) On Tuesday C) On Wednesday D) On Thursday E) On Friday Problem Kangur_2002_0506_20 (4 pts) http://www.mathkangaroo.org In the picture below, the area of triangle ABD is equal to 15, the area of triangle ABC is equal to 12 and the area of triangle ABE is equal to 4. What is the area of pentagon ABCED?
A) 19 B) 31 C) 23 D) 27 E) 35 Problem Kangur_2002_0506_21 (5 pts) http://www.mathkangaroo.org The weight of each possible pair of boys from a group of 5 was recorded. The following results were obtained: 90 kg, 92 kg, 93 kg, 94 kg, 95 kg, 96 kg, 97 kg, 98 kg, 100 kg and 101 kg. What is the total weight of all five boys?
A) 225 kg B) 230 kg C) 239 kg D) 240 kg E) 250 kg Problem Kangur_2002_0506_22 (5 pts) http://www.mathkangaroo.org There are four congruent squares. In each of them the midpoints of the sides are indicated and some regions with areas S1, S2, S3 and S4 are shaded. Which expression below is true?
A) S3 < S4 < S1 = S2 B) S3 < S1= S2 = S4 C) S3 < S1 = S4 < S2 D) S3 < S4 < S1 < S2 E) S4 < S3 < S1 < S2 Problem Kangur_2002_0506_23 (5 pts) http://www.mathkangaroo.org You count from 1 to 100 and you clap when you say the multiples of number 3 and the numbers that are not multiples of 3 but have 3 as the last digit. How many times will you clap your hands? A) 30 B) 33 C) 36 D) 39 E) 43 Problem Kangur_2002_0506_24 (5 pts) http://www.mathkangaroo.org The cyclist went up the hill with the speed of 12 km/h and went down the hill with the speed of 20 km/h. The ride up the hill took him 16 minutes longer than the ride down the hill. How many minutes did the cyclist take to go down the hill? A) 24 B) 40 C) 32 D) 16 E) 28 Problem Kangur_2002_0506_25 (5 pts) http://www.mathkangaroo.org
Symbols P, Q, R, S indicate the total weight of the figures drawn above them.
It is known that any two figures of the same shape have the same weight. If P < Q < R, then: A) P < S < Q B) Q < S < R C) S < P D) R < S E) R = S Problem Kangur_2002_0506_26 (5 pts) http://www.mathkangaroo.org Ada has 14 gray balls, 8 white balls and 6 black balls in a bag. What is the least number of the balls she has to take out of her bag having her eyes closed to make sure that she took at least one ball of each color? A) 23 B) 22 C) 21 D) 15 E) 9 A Problem Kangur_2002_0506_27 (5 pts) http://www.mathkangaroo.org
A computer virus destroys computer memory. On the first day it destroyed memory. On the second day it destroyed the third day it destroyed
of this
of the memory remaining after the first day; on
of the memory remaining after two days and on the fourth day it
destroyed of the memory remaining after three days. What part of all the computer memory was left after those four days?
A) B) C) D) E)
Problem Kangur_2002_0506_28 (5 pts) http://www.mathkangaroo.org What is the greatest value of the sum of the digits of the number made from the sum of the digits of a three-digit number? A) 9 B) 10 C) 11 D) 12 E) 18 Problem Kangur_2002_0506_29 (5 pts) http://www.mathkangaroo.org In the chess competition 32 players were competing. The competition was taking place by steps. In each step all the players were divided into groups of four. In each of these groups every player played once with every other player. The two best players from the group went to the next level and the two worst players were out of the competition. After the step in which four last players played, the two best players were playing an additional final game. How many games were played during the whole competition? A) 49 B) 89 C) 91 D) 97 E) 181 Problem Kangur_2002_0506_30 (5 pts) http://www.mathkangaroo.org
A net with 32 hexagonal spaces in three rows was made out of matches (see the picture.) How many matches were used to make this net? A) 123 B) 124 C) 125 D) 120 E) 121 Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Grade 07-08 Problem Kangur_2005_0708_1 (3 pts) http://www.mathkangaroo.org 2005 · 5002 = A) 1291 B) 102910 C) 10029010 D) 1000290010 E) 100002900010 ( · denotes multiplication) Problem Kangur_2005_0708_2 (3 pts) http://www.mathkangaroo.org How many hours are there in half of a third part of a quarter of a day?
A) B) C) 1 D) 2 E) 3 Problem Kangur_2005_0708_3 (3 pts) http://www.mathkangaroo.org The edge of the cube is 12 cm long. The ant moves on the cube surface from point A to point B along the path shown in the figure. Find the length of the ant's path.
A) 60cm B) 50cm C) 48cm D) 40cm E) It cannot be determined. Problem Kangur_2005_0708_4 (3 pts) http://www.mathkangaroo.org The sum of the volume of three pitchers and two bottles equals 16 liters. The volume of each pitcher is two times greater than the volume of each bottle. What is the sum of the volume of two pitchers and three bottles? A) 12 liters B) 13 liters
C) 14 liters D) 16 liters E) 17 liters Problem Kangur_2005_0708_5 (3 pts) http://www.mathkangaroo.org At our school, 50% of the students have bikes. Of the students who have bikes, 30% have skateboards. What percent of the students at our school have both a bike and a skateboard? A) 15 B) 20 C) 25 D) 40 E) 80 Problem Kangur_2005_0708_6 (3 pts) http://www.mathkangaroo.org In triangle ABC, the measure of the angle at vertex A is three times the measure of the angle at vertex B and half the measure of the angle at vertex C. What is the measure of the angle at vertex A? A) 30º B) 36º C) 54º D) 60º E) 72º Problem Kangur_2005_0708_7 (3 pts) http://www.mathkangaroo.org How many three-digit numbers are there in which all the digits are even? A) 25 B) 64 C) 75 D) 100 E) 125 Problem Kangur_2005_0708_8 (3 pts) http://www.mathkangaroo.org The diagram shows the floor plan of a room. Adjacent walls are perpendicular to each other. Letters a and b represent the lengths of some the walls. What is the area of the room?
A) 2ab + a(b-a) B) 3a(a+b) - a² C) 3a²b D) 3a(b-a) + a² E) 3ab Problem Kangur_2005_0708_9 (3 pts) http://www.mathkangaroo.org In the diagram, the five circles have the same radii, and they touch as shown. The small square joins the centres of the four outer circles. What is the ratio of the shaded area of all the circles to the non-shaded area of all the circles?
A) 2 : 3 B) 1 : 3 C) 2 : 5 D) 5 : 4 E) 1 : 4 Problem Kangur_2005_0708_10 (3 pts) http://www.mathkangaroo.org There was a certain number of crows sitting on trees in the garden. If there had been just one crow sitting on each tree, then one crow would not have had a tree to sit on. However, if two crows had been sitting on each tree, then there would not be any crows on one tree. How many trees were there in the garden? A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2005_0708_11 (4 pts) http://www.mathkangaroo.org Anna, Barbara, Teddy, and Wally went to a dance. They danced in pairs. Anna danced with Teddy and with Wally. Barbara danced with Teddy, but she didn't dance with Wally. Decide which statement is false. A) Each of the girls danced with one of the two boys. B) One of the two girls didn't dance with one of the two boys. C) One of the boys danced with both girls. D) Each of the boys danced with one of the two girls. E) One of the boys didn't dance with any of the two girls.
Problem Kangur_2005_0708_12 (4 pts) http://www.mathkangaroo.org A group of classmates was planning a trip. If each of them paid $14, then they would be $4 short to pay for the trip. On the other hand, if each of them paid $16, they would have $6 more than they needed. How much should each of the classmates contribute so they collect the exact amount needed for the trip? A) $14.40 B) $14.60 C) $14.80 D) $15.00 E) $15.20 Problem Kangur_2005_0708_13 (4 pts) http://www.mathkangaroo.org Carla cut a sheet of paper into 10 pieces. Then she took one piece and cut it again into 10 pieces. She then repeated this three more times. How many pieces of paper did she have after the last cutting? A) 36 B) 40 C) 46 D) 50 E) 56 Problem Kangur_2005_0708_14 (4 pts) http://www.mathkangaroo.org A doorman works according the following schedule: he works for 4 consecutive days and has the fifth day off. Last Sunday he had the day off, and on Monday he started work according to his schedule. After how many days, including that Monday, will he have a day off on Sunday again? A) 30 B) 36 C) 12 D) 34 E) 7 Problem Kangur_2005_0708_15 (4 pts) http://www.mathkangaroo.org Two rectangles ABCD and DBEF are shown in the picture. What is the area of rectangle DBEF?
A) 10 cm² B) 12 cm² C) 13 cm² D) 14 cm² E) 16 cm² Problem Kangur_2005_0708_16 (4 pts) http://www.mathkangaroo.org What is the measure of angle indicated in the picture?
A) 110º B) 115º C) 120º D) 126º E) 130º Problem Kangur_2005_0708_17 (4 pts) http://www.mathkangaroo.org From noon until midnight, Clever Cat sleeps under the oak tree, and from midnight until noon he tells stories. There is a sign on the oak tree saying: "Two hours ago Clever Cat was doing the same thing that he will be doing in an hour." How many hours a day is the information given on the sign true? A) 6 B) 12 C) 18 D) 3 E) 21 Problem Kangur_2005_0708_18 (4 pts) http://www.mathkangaroo.org The diagram shows an equilateral triangle and a regular pentagon. What is the measure of angle x?
A) 124º B) 128º C) 132º
D) 136º E) 140º Problem Kangur_2005_0708_19 (4 pts) http://www.mathkangaroo.org Which of the numbers below is the sum of four consecutive whole numbers? A) 15 B) 2000 C) 2002 D) 2004 E) 2005 Problem Kangur_2005_0708_20 (4 pts) http://www.mathkangaroo.org Each pair vertices of a cube are connected with a segment. How many different points are there that are the midpoints of these segments? A) 8 B) 12 C) 16 D) 19 E) 28 Problem Kangur_2005_0708_21 (5 pts) http://www.mathkangaroo.org Let's call the number of prime factors of a natural number n, the product of which is equal to the given natural number n, "the length". For example, the length of 90 = 2x3x3x5 equals 4. How many odd numbers less than 100 have the length of 3? A) 2 B) 3 C) 5 D) 7 E) Other number. Problem Kangur_2005_0708_22 (5 pts) http://www.mathkangaroo.org In each of the four small squares shown in the picture, a different natural odd number less than 20 was written. Only one of the statements below is true. Which one is it?
A)The sum of the numbers that are inscribed in the square equals 66. B)The sum of the numbers that are inscribed in the square equals 12. C)Product of all the numbers inscribed in the square equals 2005.
D)The product of the numbers on each diagonal equals 21. E)The sum of the number on each diagonal equals 20. Problem Kangur_2005_0708_23 (5 pts) http://www.mathkangaroo.org The sequence of letters AGKNORU in alphabetical order corresponds to a sequence of different digits placed in increasing order. What is the greatest number that corresponds with the word KANGOUROU? A) 987654321 B) 987654354 C) 436479879 D) 536479879 E) 597354354 Problem Kangur_2005_0708_24 (5 pts) http://www.mathkangaroo.org Peter wrote down all three-digit numbers which have the following features: each number consists of three different digits, and the first digit is equal to the second power of the quotient of the second and the third digit. How many numbers did Peter write down? A) 8 B) 4 C) 3 D) 2 E) 1 Problem Kangur_2005_0708_25 (5 pts) http://www.mathkangaroo.org Five lines: l1, l2, l3, l4, l5 intersect point O and they are intersected by five other lines k1, k2, k3, k4, k5 (see the picture). What is the sum of the measures of 10 shaded angles, shown in the picture?
A) 300 B) 450
C) 360 D) 600 E) 720 Problem Kangur_2005_0708_26 (5 pts) http://www.mathkangaroo.org There were 64 liters of juice in a barrel. Then, 16 liters of juice were dumped out and replaced with 16 liters of water. After being mixed together, again 16 liters of the mixture were dumped and replaced with 16 liters of water. After being mixed together, this was done again, 16 liters of the mixture were dumped and replaced with water. How many liters of juice are there in the mixture now? A) 27 B) 24 C) 16 D) 30 E) 48 Problem Kangur_2005_0708_27 (5 pts) http://www.mathkangaroo.org How many two-digit numbers with the ones digit greater than zero are there that are greater than three times the number that is created out of these numbers by reversing its digits? A) 6 B) 10 C) 15 D) 22 E) 33 Problem Kangur_2005_0708_28 (5 pts) http://www.mathkangaroo.org ABC is a right triangle. AH is the height of the triangle and AK is a bisector of right angle A. If the ratio CK : KB equals
then what is the ratio of CH : HB?
A) 1 : 3 B) 1 : 9 C) 1 : D) 1 : 6 E) 1 : 4 Problem Kangur_2005_0708_29 (5 pts) http://www.mathkangaroo.org
If the average of 10 different positive integers is 10, how large can the greatest of these numbers be? A) 91 B) 55 C) 50 D) 45 E) 10 Problem Kangur_2005_0708_30 (5 pts) http://www.mathkangaroo.org A particle moves through the first quadrant of the figure as follows: during the first minute it moves from the origin to (1,0). Then, it continues to follow the pattern indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in one minute. Which point will the particle reach after exactly 2 hours?
A) (10,0) B) (1,11) C) (10,11) D) (2,10) E) (11,11) Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2004_0708_1 (3 pts) http://www.mathkangaroo.org What is the value of the expression: 2004 - 200 · 4? A) 400,800 B) 0 C) 1204 D) 1200 E) 2804 Problem Kangur_2004_0708_2 (3 pts) http://www.mathkangaroo.org Tom has $147 and Stan has $57. How much money does Tom need to give to Stan, so that he would have twice as much money left as Stan would have then?
A) $11 B) $19 C) $30 D) $45 E) $49 Problem Kangur_2004_0708_3 (3 pts) http://www.mathkangaroo.org What is the remainder when dividing the sum: 2001 + 2002 + 2003 + 2004 + 2005 by 2004? A) 1 B) 2001 C) 2002 D) 2003 E) 1999 Problem Kangur_2004_0708_4 (3 pts) http://www.mathkangaroo.org In each of the little squares Karolina places one of the digits: 1, 2, 3, 4. She makes sure that in each row and each column each of these numbers is placed. In the figure below, you can see the way she started. In how many ways can she fill the square marked with an x? 1
x
4 1 3 2 A) None B) 1 C) 2 D) 3 E) 4 Problem Kangur_2004_0708_5 (3 pts) http://www.mathkangaroo.org What is the value of the expression: (1 - 2) - (3 - 4) - (5 - 6) - (7 - 8) - (9 - 10) - (11 - 12)? A) -6 B) 0 C) 4 D) 6 E) 13 Problem Kangur_2004_0708_6 (3 pts) http://www.mathkangaroo.org
A section was made on a cube. On the net of the cube this section was indicated with a perforated line (see the figure). What figure was made by the section?
A) Equilateral triangle B) A rectangle but not a square C) Right triangle D) Square E) Hexagon Problem Kangur_2004_0708_7 (3 pts) http://www.mathkangaroo.org By how much does the area of a rectagle increase if its length and the width are increased by 10% each? A) 10% B) 20% C) 21% D) 100% E) 121% Problem Kangur_2004_0708_8 (3 pts) http://www.mathkangaroo.org What is the length of the diameter of the circle shown in the figure?
A) 18 B) 16 C) 10 D) 12 E) 14 Problem Kangur_2004_0708_9 (3 pts) http://www.mathkangaroo.org An ice cream stand was selling ice cream in five different flavors. A group of children came to the stand and each child bought two scoops of ice cream with two different flavors. If none of the children chose the same combination of flavors and every such combination of flavors was chosen, how many children were there?
A) 5 B) 10 C) 20 D) 25 E) 30 Problem Kangur_2004_0708_10 (3 pts) http://www.mathkangaroo.org The number x was multiplied by 0.5 and the product was divided by 3. The result was squared and 1 was added to it. The final result was 50. What was the value of number x? A) 18 B) 24 C) 30 D) 40 E) 42 Problem Kangur_2004_0708_11 (4 pts) http://www.mathkangaroo.org Alfonso the ostrich was training for the Head in the Sand Competition in the Animal Olympiad. He put his head in the sand at 8:15 on Monday morning and reached his new personal record by keeping it underground for 98 hours and 56 minutes. When did Alfonso pull his head out of the sand? A) On Thursday at 5:19 A.M. B) On Thursday at 5:41 A.M. C) On Thursday at 11:11 A.M. D) On Friday at 5:19 A.M. E) On Friday at 11:11 A.M. Problem Kangur_2004_0708_12 (4 pts) http://www.mathkangaroo.org Two semicircles with diameters AB and AD were inscribed in square ABCD (see the figure). If |AB| = 2, then what is the area of the shaded region?
A) 1 B) 2 C) D) 2 E) Problem Kangur_2004_0708_13 (4 pts) http://www.mathkangaroo.org If a and b are positive integers, neither of which is divisible by 10, and if a · b = 10,000 then what is the sum a + b? A) 1024 B) 641 C) 1258 D) 2401 E) 1000 Problem Kangur_2004_0708_14 (4 pts) http://www.mathkangaroo.org There were more Thursdays than Tuesdays in the first of two consecutive years. Which day of the week appeared the most in the second year, if neither of these years was a leap year? A) Tuesday B) Wednesday C) Friday D) Saturday E) Sunday Problem Kangur_2004_0708_15 (4 pts) http://www.mathkangaroo.org Isosceles triangle ABC satisfies: |AB| = |AC| = 5, and angle BAC > 60°. The length of the perimeter of this triangle is expressed with a whole number. How many triangles of that kind are there? A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2004_0708_16 (4 pts) http://www.mathkangaroo.org How many divisors does number 2 x 3 x 5 x 7 x 11 have? A) 2310 B) 10 C) 5
D) 2004 E) 32 Problem Kangur_2004_0708_17 (4 pts) http://www.mathkangaroo.org Tad has a large number of building blocks which are rectangular prisms with dimensions 1 x 2 x 3. What is the smallest number of blocks needed to build a solid cube? A) 12 B) 18 C) 24 D) 36 E) 60 Problem Kangur_2004_0708_18 (4 pts) http://www.mathkangaroo.org Each of 5 children wrote one of the numbers: 1, 2, 4 on the board. Then the written numbers were multiplied. Which number can be the product of those numbers? A) 100 B) 120 C) 256 D) 768 E) 2048 Problem Kangur_2004_0708_19 (4 pts) http://www.mathkangaroo.org The average age of a grandmother, a grandfather and 7 grandchildren is 28. The average age of 7 grandchildren is 15 years. How old is the grandfather, if he is 3 years older than the grandmother? A) 71 B) 72 C) 73 D) 74 E) 75 Problem Kangur_2004_0708_20 (4 pts) http://www.mathkangaroo.org The equilateral triangle ACD is rotated counterclockwise around point A. What is the angle of rotation when triangle ACD covers triangle ABC the first time?
A) 60° B) 120° C) 180° D) 240° E) 300° Problem Kangur_2004_0708_21 (5 pts) http://www.mathkangaroo.org There are at least two kangaroos in the enclosure. One of them said: "There are 6 of us here" and he jumped out of the enclosure. Afterwards, every minute one kangaroo was jumping out of the enclosure saying: "Everybody who jumped out before me was lying." This continued until there were no kangaroos left in the enclosure. How many kangaroos were telling the truth? A) 0 B) 1 C) 2 D) 3 E) 4 Problem Kangur_2004_0708_22 (5 pts) http://www.mathkangaroo.org Points A and B are placed on a line which connects the midpoints of two opposite sides of a square with side of 6 cm (see the figure). When you draw lines from A and B to two opposite vertices, you divide the square in three parts of equal area. What is the length of segment AB?
A) 3.6 cm B) 3.8 cm C) 4.0 cm D) 4.2 cm E) 4.4 cm Problem Kangur_2004_0708_23 (5 pts) http://www.mathkangaroo.org Jack rides his bike from home to school uphill with average speed of 10 km/h. On the way back home his speed is 30km/h. What is the average speed of his round trip? A) 12 km/h B) 15 km/h C) 20 km/h D) 22 km/h E) 25km/h
Problem Kangur_2004_0708_24 (5 pts) http://www.mathkangaroo.org John put magazines on a bookshelf. They have either 48 or 52 pages. Which one of the following numbers cannot be the total number of pages of all the magazines on the bookshelf? A) 500 B) 524 C) 568 D) 588 E) 620 Problem Kangur_2004_0708_25 (5 pts) http://www.mathkangaroo.org Inside the little squares of a big square the consecutive natural numbers were placed in a way shown in the picture. Which of the following numbers cannot be placed in square x?
A) 128 B) 256 C) 81 D) 121 E) 400 Problem Kangur_2004_0708_26 (5 pts) http://www.mathkangaroo.org In the figure there are 11 boxes. Number 7 was written in the first box and number 6 was written in the ninth box. What was the number placed in the second field with the following condition: the sums of each three consecutive numbers in the boxes are equal to 21?
A) 7 B) 10 C) 8 D) 6 E) 21 Problem Kangur_2004_0708_27 (5 pts) http://www.mathkangaroo.org For each triple of numbers (a, b, c) another triple of numbers (b + c, c + a, a + b) was created. This was called operation. 2004 such operations were made starting with numbers (1, 3, 5), and resulting with numbers (x, y, z). What is the difference x - y equal to?
A) -2 B) 2 C) 4008 D) 2004 E) (-2)2004 Problem Kangur_2004_0708_28 (5 pts) http://www.mathkangaroo.org Number 2004 is divisible by 12 and the sum of its digits is equal to 6. Altogether, how many four-digit numbers have these two properties? A) 10 B) 12 C) 13 D) 15 E) 18 Problem Kangur_2004_0708_29 (5 pts) http://www.mathkangaroo.org Rings with dimensions shown in the figure were linked together, forming 1.7m long chain. How many rings were used to create the chain?
A) 30 B) 21 C) 42 D) 85 E) 17 Problem Kangur_2004_0708_30 (5 pts) http://www.mathkangaroo.org On each face of a cube a certain natural number was written, and at each vertex a number equal to the product of the numbers on the three faces adjacent to that vertex was placed. If the sum of the numbers on the vertices is 70 then what is the sum of the numbers on all the faces of the cube? A) 12 B) 35 C) 14 D) 10 E) Cannot be determined. Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2003_0708_1 (3 pts) http://www.mathkangaroo.org A segment of a length equal to 4 was divided with 4 points into segments of equal length. How long is each segment? A) 0.4 B) 1 C) 0.8 D) 0.5 E) 0.6 Problem Kangur_2003_0708_2 (3 pts) http://www.mathkangaroo.org A square built of 16 small squares is cut with a line. What is the greatest number of the little squares that the line can go through?
A) 3 B) 4 C) 6 D) 7 E) 8 Problem Kangur_2003_0708_3 (3 pts) http://www.mathkangaroo.org When 29 is subtracted from the greatest 2-digit number and the difference is divided by the smallest 2-digit number what is the result? A) 11 B) 9 C) 7 D) 10 E) 6 Problem Kangur_2003_0708_4 (3 pts) http://www.mathkangaroo.org
If A) 9 B) 2 C) 10 D) 3 E) 15
, then
is equal to?
Problem Kangur_2003_0708_5 (3 pts) http://www.mathkangaroo.org Tomek saved 120 zl. One day he bought a present for his brother and he spent 1/3 of all his money. The next day he bought a book for himself and he spent 1/4 of the remaining money. How much money did he have left after shopping? (zl is a monetary unit in Poland like dollar in USA) A) 50 zl B) 80 zl C) 70 zl D) 20 zl E) 60 zl Problem Kangur_2003_0708_6 (3 pts) http://www.mathkangaroo.org A rectangular prism was made out of three blocks, each consisting of four cubes (see the picture). Which of the blocks below has the same shape as the white block?
Problem Kangur_2003_0708_7 (3 pts) http://www.mathkangaroo.org There are 17 trees on one side of the street on Tomek's way from his house to school. One day Tomek marked these trees with white chalk in the following way: on the way from his house to the school he marked every other tree, starting with the first one. On his way back home he marked every third tree, starting with the first one. How many trees were not marked? A) 4 B) 5 C) 6 D) 7 E) 8 Problem Kangur_2003_0708_8 (3 pts) http://www.mathkangaroo.org In triangle ABC : DA = DB = DC (see the figure). Then:
A) Angle ACB is obtuse. B) Angle ACB is acute. C) Angle ACB is right. D) Angle ACB changes, depending on the lengths of the sides AC and BC. E) This kind of triangle ABC does not exist. Problem Kangur_2003_0708_9 (3 pts) http://www.mathkangaroo.org There were 5 parrots in a pet store. The average price of each of them was 600 zl. One day, the most beautiful parrot was sold. The average price of each of the remaining 4 birds was 500 zl.What was the price of the sold parrot? (zl is a monetary unit in Poland like dollar in USA) A) 100 zl B) 200 zl C) 550 zl D) 600 zl E) 1000 zl Problem Kangur_2003_0708_10 (3 pts) http://www.mathkangaroo.org What is the greatest number of inner right angles that a hexagon can have (not necessary a convex hexagon)? A) 2 B) 3 C) 4 D) 5 E) 6 Problem Kangur_2003_0708_11 (4 pts) http://www.mathkangaroo.org How many integers are there, the squares of which are found between the numbers -100 and 100, inclusive? A) 11 B) 22 C) 21 D) 20 E) 10 Problem Kangur_2003_0708_12 (4 pts) http://www.mathkangaroo.org Five girls Ania, Beata, Celina, Dorota, and Ela, received the following assignment: they were supposed to draw four segments and to count all the points of intersection of these segments. After they finished, Ania said she counted 2 points, Beata 3 points, Celina 5 points, Dorota 6 points, Ela 7 points. Who made a mistake? A) Ania B) Beata
C) Celina D) Dorota E) Ela Problem Kangur_2003_0708_13 (4 pts) http://www.mathkangaroo.org A cube was made from the given configuration (see the figure). Which wall will be opposite to the wall with the letter x?
A) a B) b C) c D) d E) e Problem Kangur_2003_0708_14 (4 pts) http://www.mathkangaroo.org A square piece of paper is folded twice and cut in the way you can see in the picture. How will the piece of paper look after unfolding?
Problem Kangur_2003_0708_15 (4 pts) http://www.mathkangaroo.org In how many ways can you express the number 2003 as a sum of two prime numbers? A) 1 B) 2 C) 3 D) 4 E) This expression doesn't exist
Problem Kangur_2003_0708_16 (4 pts) http://www.mathkangaroo.org Which of the following sums is equal to 2003? A)162 + 262 + 322 B)162 + 262 + 332 C)152 + 272 + 322 D)172 + 252 + 332 E)172 + 242 + 342 Problem Kangur_2003_0708_17 (4 pts) http://www.mathkangaroo.org An empty truck weighs 2000 kg. After the truck was loaded, freight made up 80% of the weight of the loaded truck. At the first stop one fourth of the freight was unloaded. What percent of the loaded truck's weight did the load make up after that? (Hint: freight = load.) A) 20% B) 25% C) 55% D) 60% E) 75% Problem Kangur_2003_0708_18 (4 pts) http://www.mathkangaroo.org The combined capacity of a bottle and a glass is equal to the capacity of a pitcher. The capacity of a bottle is equal to the combined capacity of a glass and a mug. The combined capacity of three mugs is equal to the combined capacity of two pitchers. How many glasses altogether have the capacity of one mug? A) 3 B) 4 C) 5 D) 6 E) 7 Problem Kangur_2003_0708_19 (4 pts) http://www.mathkangaroo.org Michal has 42 identical cubical blocks, each one with an edge of 1 cm. From all of these blocks, he built a rectangular prism with a base perimeter equal to 18 cm. What is the height of the prism which he built? A) 1 cm B) 2 cm C) 3 cm D) 4 cm E) 5 cm Problem Kangur_2003_0708_20 (4 pts) http://www.mathkangaroo.org
A fresh mushroom contains 90% water. How many kilograms of fresh mushrooms are needed in order to make 1 kg of dried mushrooms containing 11% water? A) 8.9 B) 79 C) 1.01 D) 8.18 E) 9.09 Problem Kangur_2003_0708_21 (5 pts) http://www.mathkangaroo.org There are five men in a room. Each of them is either a liar that always lies or a knight that always tells the truth. Each of them was asked the question: "How many liars are among you?" The answers were: "one," "two," "three," "four," "five." How many liars were in that room? A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2003_0708_22 (5 pts) http://www.mathkangaroo.org In rectangle ABCD, points P, Q, R, S are respectively the midpoints of sides AB, BC, CD, and DA. Let T be the midpoint of segment SR. What part of the area of rectangle ABCD is the area of triangle PQT?
A) B) C) D) E) Problem Kangur_2003_0708_23 (5 pts) http://www.mathkangaroo.org
There are six segments with lengths: 1, 2, 3, 2001, 2002, 2003. In how many ways can we select three of these segments to build a triangle? A) 1 B) 3 C) 5 D) 6 E) 10 Problem Kangur_2003_0708_24 (5 pts) http://www.mathkangaroo.org Six consecutive points were indicated on a number line in this order: A, B, C, D, E, and F. Regardless of the location of these points, if only AD = CF and BD = DF, then the following equation is true: A) AB = BC B) BC = DE C) BD = EF D) AB = CD E) CD = EF Problem Kangur_2003_0708_25 (5 pts) http://www.mathkangaroo.org In the picture four squares are shown and the lengths of their sides are indicated. What is the difference between the combined area of the shaded regions and the combined area of the black regions?
A) 25 B) 36 C) 44 D) 64 E) 0 Problem Kangur_2003_0708_26 (5 pts) http://www.mathkangaroo.org Which of the following numbers, after it is multiplied by 768, yields a product that ends with the largest number of zeros? A) 7,500 B) 5,000 C) 3,125
D) 2,500 E) 10,000 Problem Kangur_2003_0708_27 (5 pts) http://www.mathkangaroo.org A square was divided into 25 identical small squares (see the picture). What is the sum of the measures of angles < MAN, < MBN, < MCN, < MDN, < MEN?
A) 300 B) 450 C) 600 D) 750 E) 900 Problem Kangur_2003_0708_28 (5 pts) http://www.mathkangaroo.org How many natural numbers n have such a property that out of all the positive divisors of number n, which are different from both 1 and n, the greatest one is 15 times greater than the smallest one? A) 1 B) 2 C) 3 D) There are no such numbers. E) Infinitely many. Problem Kangur_2003_0708_29 (5 pts) http://www.mathkangaroo.org In a number with at least two digits, the last digit was deleted. The resulting number was n times smaller than the previous one. What is the greatest possible value of n? A) 9 B) 10 C) 11 D) 19 E) 20 Problem Kangur_2003_0708_30 (5 pts) http://www.mathkangaroo.org
How many natural numbers n have the property that the remainder of dividing 2003 by n is equal to 23? A) 22 B) 19 C) 13 D) 12 E) 36 Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
Problem Kangur_2002_0708_1 (3 pts) http://www.mathkangaroo.org This year the International Competition in Mathematics "Kangaroo" takes places on March 21st. How many prime numbers divide the number 21? A) 2 B) 3 C) 4 D) 1 E) 21 Problem Kangur_2002_0708_2 (3 pts) http://www.mathkangaroo.org Which of the fractions below is the greatest?
A)
B)
C)
D)
E)
Problem Kangur_2002_0708_3 (3 pts) http://www.mathkangaroo.org You count from 1 to 100 and you clap while saying the multiples of the number 3 and the numbers that are not the multiples of three but have 3 as the last digit. How many times will you clap your hands? A) 30 B) 33 C) 36 D) 39 E) 43 Problem Kangur_2002_0708_4 (3 pts) http://www.mathkangaroo.org On July 1st in Newbury the sun will rise at 4:53 A.M. and set at 9:25 P.M. In the middle of that period of time there is so called local noon. At what time will the local noon be in Newbury on July 1st?
A) At 12:00 P.M. B) At 12:39 P.M. C) At 1:09 P.M. D) At 4:32 P.M. E) At 11:08 A.M. Problem Kangur_2002_0708_5 (3 pts) http://www.mathkangaroo.org Points K, L, M, N are the midpoints of the sides of rectangle ABCD and points O, P, R, S are the midpoints of the sides of rhombus KLMN. What is the ratio of the area of shaded figure to the area of rectangle ABCD?
A)
B)
C)
D)
E)
Problem Kangur_2002_0708_6 (3 pts) http://www.mathkangaroo.org Ada has 7 gray balls, 4 white balls and 3 black balls in a bag. What is the least number of the balls she has to take out of her bag, having her eyes covered, to make sure that she took out at least one ball of each color? A) 12 B) 11 C) 10 D) 4 E) 3 Problem Kangur_2002_0708_7 (3 pts) http://www.mathkangaroo.org A certain charity organization decided to buy 2002 notebooks. The warehouse was selling boxes of 24 notebooks. What is the least number of boxes that the warehouse should buy in order to have 2002 notebooks, and by what number will the number of 2002 notebooks be exceeded? A) 83 boxes, 10 notebooks B) 84 boxes, 10 notebooks C) 83 boxes, 14 notebooks D) 84 boxes, 16 notebooks E) 84 boxes, 14 notebooks Problem Kangur_2002_0708_8 (3 pts) http://www.mathkangaroo.org Which of the expressions below cannot have the value of 2002, if a and b represent the natural numbers?
A) 7a + 7b B) 13a + 13b C) 17a + 17b D) 11(2a + 7b) E) 28a + 14b Problem Kangur_2002_0708_9 (3 pts) http://www.mathkangaroo.org Each boy: Mietek, Mirek, Pawel and Zbyszek has exactly one of four animals: a cat, a dog, a gold fish and a canary. Mirek has an animal with fur, Zbyszek has an animal with four legs, Pawel has a bird and Mietek and Mirek don't like cats. Which of the statements below is not true? A) Zbyszek has a dog B) Pawel has a canary C) Mietek has a gold fish D) Zbyszek has a cat E) Mirek has a dog Problem Kangur_2002_0708_10 (3 pts) http://www.mathkangaroo.org A basket of oranges costs 20 zloty, a basket of pears costs 30 zloty and a basket of kiwi fruits costs 40 zloty. Eight baskets of these fruits were bought for 230 zloty. What is the largest possible number of baskets of kiwi fruits that were bought? A) 1 B) 2 C) 3 D) 4 E) 5 Problem Kangur_2002_0708_11 (4 pts) http://www.mathkangaroo.org If a : b = 9 : 4 and b : c = 5 : 3 then (a - b) : (b - c) is equal to: A) 4 : 1 B) 25 :8 C) 7:12 D) 5 : 2 E) It cannot be determined Problem Kangur_2002_0708_12 (4 pts) http://www.mathkangaroo.org Before going to a summer camp, the scouts from Torun packed provisions that were sufficient for them for 30 days. At the last minute 15 scouts from Bydgoszcz wanted to go to the summer camp with the scouts from Torun. Now the provisions already made were sufficient for just 25 days provided the daily amount of food allotted for one scout would not change. How many scouts from Torun were planning to go for that summer camp? A) 15 B) 20
C) 55 D) 70 E) 75 Problem Kangur_2002_0708_13 (4 pts) http://www.mathkangaroo.org There were 25% boys and 75% girls among all the students taking part in the school event. Half of the boys and 20% of the girls, together 99 students, had blue eyes. How many students were taking part in the school event? A) 360 B) 340 C) 240 D) Other answer E) It cannot be determined Problem Kangur_2002_0708_14 (4 pts) http://www.mathkangaroo.org Points P and Q are the centers of two outside tangent circles (see the picture.) The line going through points P and Q intersects these circles at points A and B. If the area of rectangle ABCD is 15 then what is the area of triangle PQT?
A) 4 B)
C)
D) 5 E) 2
Problem Kangur_2002_0708_15 (4 pts) http://www.mathkangaroo.org The weight of each possible pair of boys from a group of 5 was recorded. The following results were obtained: 90 kg, 92 kg, 93 kg, 94 kg, 95 kg, 96 kg, 97 kg, 98 kg, 100 kg and 101 kg. What is the total weight of the five boys? A) 225 kg B) 230 kg C) 239 kg D) 240 kg E) 250 kg Problem Kangur_2002_0708_16 (4 pts) http://www.mathkangaroo.org Four daughters, Ania, Basia, Celina and Danusia, bought together a single present for their dad. One of the girls hid the present. Mom asked which of them had done it. Ania and Basia said: "I did not do it." Celina: "Danusia did it." Danusia: "Basia did it." It turned out that only one of the girls lied. Who of them hid the present?
A) Ania B) Basia C) Celina D) Danusia E) It cannot be determined Problem Kangur_2002_0708_17 (4 pts) http://www.mathkangaroo.org In one country a part of the residents can speak English only, a part can speak French only and the rest can speak both languages. It is known that 85% residents can speak English, 75% can speak French. What percent of the residents of this country can speak both English and French? A) 50% B) 57% C) 25% D) 60% E) 40% Problem Kangur_2002_0708_18 (4 pts) http://www.mathkangaroo.org The symbols P, Q, R, S indicate the total weight of the figures drawn above them (see the picture):
It is known that any two figures of the same shape have the same weight. If P < Q < R then: A) P < S < Q B) Q < S < R C) S
A) 90° B) 18° C) 36°
D) 54° E) 72° Problem Kangur_2002_0708_20 (4 pts) http://www.mathkangaroo.org Out of the net shown in the picture the cube was made. What is the greatest sum of the dots on three sides with a common vertex?
A) 15 B) 14 C) 13 D) 12 E) Other answer Problem Kangur_2002_0708_21 (5 pts) http://www.mathkangaroo.org It is known that the positive whole number n is divisible by 21 and by 9. Which of the answers below can be the number of divisors of the number n? A) 3 B) 4 C) 5 D) 6 E) 7 Problem Kangur_2002_0708_22 (5 pts) http://www.mathkangaroo.org In some of the segments of the rectangular diagram with the dimensions 2 x 9 (see the picture below) there are coins. Each of the segment either has a coin in it or has a side common with the segment containing a coin. What is the least number of coins that must be in this diagram?
A) 5 B) 6 C) 7 D) 8 E) 9 Problem Kangur_2002_0708_23 (5 pts) http://www.mathkangaroo.org In one month three Sundays were on even dates. What day of the week was the 20th day of the month?
A) Monday B) Tuesday C) Wednesday D) Thursday E) Saturday Problem Kangur_2002_0708_24 (5 pts) http://www.mathkangaroo.org The front face of the clock cracked into three parts in such a way that in each part the sum of the numbers indicating the hours was the same. Knowing that none of the lines along which the crack happened divides the digits of the number we can say that:
A) 12 and 3 are not in the same part B) 8 and 4 are in the same part C) 7 and 5 are not in the same part D) 11, 1 and 5 are in the same part E) 2, 11 and 9 are in the same part Problem Kangur_2002_0708_25 (5 pts) http://www.mathkangaroo.org Following the teacher's assignment, the students were drawing two circles and three lines on the piece of paper. After that each of them was counting the points of the intersection of these lines. What is the biggest possible number of such points? A) 18 B) 17 C) 16 D) 15 E) 14 Problem Kangur_2002_0708_26 (5 pts) http://www.mathkangaroo.org A square piece of paper was folded into a pentagon in the following way: first we fold the square in a way so that the vertices B and D would go into one point laying on the diagonal AC (see picture 2), and then we fold the resulting quadrilateral in a way so that point C would go into point A (see picture 3.) What is the measure of the angle with the question mark?
Picture 1 ... Picture 2 ... Picture 3
A) 104° B) 106° 30' C) 108° D) 112° 30' E) 114° 30' Problem Kangur_2002_0708_27 (5 pts) http://www.mathkangaroo.org A solid was made out of 112 identical cubes. The solid is a cube with three tunnels drilled through it as you can see in the picture. After the glue dried the solid was dipped into a dish with paint. How many little cubes have exactly one side painted?
A) 30 B) 26 C) 40 D) 48 E) 24 Problem Kangur_2002_0708_28 (5 pts) http://www.mathkangaroo.org With the digits 1, 2, 3, 4 all possible four-digit numbers with all different digits were made. What is the sum of all these numbers? A) 55,550 B) 99,990 C) 66,660 D) 100,000 E) 98,760 Problem Kangur_2002_0708_29 (5 pts) http://www.mathkangaroo.org In the picture DC = AC = 1 and CB = CE = 4. If the area of triangle ABC is equal to S then what is the area of the quadrilateral AFDC?
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Problem Kangur_2002_0708_30 (5 pts) http://www.mathkangaroo.org During math class the teacher wrote number 1 on the board and asked Tomek to write down any other natural number. Then other students were coming up to the board and each of them wrote a number that was the sum of all the numbers written before. At some point Piotr wrote the number 1000. Which of the numbers below could not Tomek write? A) 999 B) 499 C) 299 D) 249 E) 124 Math Kangaroo in USA NFP, Inc. ® Since 2003, All Rights Reserved ©
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MONDAY, MARCH 2, 2009 Week 4 - Problem 4
Figures I, II, III and IV are squares. The circumference of square I is 16m and the circumference of square II is 24m. Find the circumference of square IV. A. 56m B. 60m C. 64m D. 72m E. 80m Week 4 - Problem 3 The area of a rectangle equals 1. What is the area of the triangle, which is cut off from the rectangle by the line connecting the midpoints of the two adjacent sides? A. 1/3 B. 1/4 C. 2/5 D. 3/8 E. 1/8 SATURDAY, FEBRUARY 28, 2009 Week 4 - Problem 2 The combination for opening a safe is a three – digit number made up of different digits. How many different combinations can you make using only digits 1, 3, and 5? A) 2 B) 3 C) 4 D) 5 E) 6 Week 4 - Problem 1 The numbers 34 and 142 have the same sum of their digits (3+4=7 and 1+4+2=7). What is the first number greater than 2007 such that the sum of its digits is the same as the sum of the digits of 2007? A) 2016 B) 2115 C) 2008 D) 7002 E) 2070 MONDAY, FEBRUARY 23, 2009 Week 3 - Problem 6 The fraction equals to a) 2003 b) 1/3 c) 3 d) 5/2 e) None Week 3 - Problem 5 Betty likes calculating the sum of the digits that she sees on her digital clock (for instance, if the clock shows 21:17, then Betty gets 11). What is the biggest sum she can get if the clock is a 24-hour clock?
A) 24 B) 36 C) 19 D) 25 E) another answer SATURDAY, FEBRUARY 21, 2009 Week 3 - Problem 4 Rachel opened her math book and found that the sum of the facing pages was 243. What pages did she open to? Week 3 - Problem 3 A math student interviewed 50 fifth graders. 41 students said they liked peanut butter sandwiches, 35 liked jam sandwiches, and 30 liked both on their sandwiches. How many students like neither? THURSDAY, FEBRUARY 19, 2009 Week 3 - Problem 2 Sophie draws kangaroos: a blue one, then a green, then a red, then a black, a blue, a green, a red, a black, and so on…What color is the 29th kangaroo? A) blue B) green C) red D) black E) it’s impossible to know Week 3 - Problem 1 The sum of the numbers in each ring below should be 55. What is the value of A? A) 9 B) 10 C) 13 D) 16 E) 17
WEDNESDAY, FEBRUARY 18, 2009 Challenge Problem In the addition problem below, different letters represent different digits. What digit does A represent? AA +AA ____ CAB Challenge Problem
Mr. Chin went to a store where he spent one-half of his money and then $14 more. He then went to another store where he spent one-third of his remaining money and then $14 more. He then had no money left. How much did he have when he entered the first store? TUESDAY, FEBRUARY 17, 2009 Week 2 - Problem 6 Anna wrote a 2-digit number. Ben created a 4-digit number by copying Anna's number twice. Then Anna divided Ben's number by her number. What was the result she got? A) 100 B) 101 C) 1000 D) 1001 E) 10 Week 2 - Problem 5 Bill thought of an integer number. Nick multiplied this number either by 5 or by 6. John added either 5 or 6 to Nick’s result. Last, Andrew subtracted either 5 or 6 from John’s result. The final result obtained was 73. What was Bill’s number? A) 10 B) 11 C) 12 D) 14 E) 15 MONDAY, FEBRUARY 16, 2009 Week 2 - Problem 4 Harry Potter let an owl out at 7:30 a.m. to deliver an important message to his friend Ron. The owl delivered the envelope at 9:10 a.m. An owl flies 4 km in 10 minutes. What was the distance between Harry and Ron? A) 14 km B) 20 km C) 40 km D) 56 km E) 64 km Week 2 - Problem 3 What is the 2007th letter in the sequence KANGAROOKANGAROOKANG… ? A) K B) A C) N D) R E) O Week 2 - Problem 2 There were 60 birds on three trees. At some moment 6 birds flew away from the first tree, 8 birds flew away from the second tree, and 4 birds flew away from the third tree. After that, it turned out that the number of birds on each tree was the same. How many birds were there on the second tree in the beginning? A) 26 B) 24 C) 22 D) 21 E) 20 Week 2 - Problem 1
The square in the figure is a mini-sudoku: the numbers 1, 2, and 3 must be written in the cells so that each of them appears in each row and in each column. Harry started to fill in the square. In how many ways can he complete the task? A) 1 B) 2 C) 3 D) 4 E) 5
CHALLENGE PROBLEM ABCD represents a four-digit number. The product of its digits is 70. What is the largest fourdigit number that ABCD can represent? CHALLENGE PROBLEM A prime number is a whole number, greater than 1, that is divisible only by itself and 1. Some examples of prime numbers are 2,3,5,7,11, and 13. What is the largest prime number, P, such that 9 times P is less than 400? WEDNESDAY, FEBRUARY 11, 2009 Week 1 - Problem 5 The odometer of my car indicates 187569. All the digits of this number are different. After how many more kilometres will this happen again? A. 1 B. 21 C. 431 D. 12431 E. 13776 Week 1 - Problem 4 Robert made a tunnel using some identical cubes (fig.1). When he got bored, he rearranged the tunnel into a complete pyramid (fig.2). How many cubes from the original tunnel did he not use for the pyramid?
A. 34 B. 29 C. 22 D. 18 E. 15 Week 1 - Problem 3 From her window Karla looks at the wall of a house. There she can see the silhouette of a rectangular flag flying in the wind. At five different moments she draws the silhouette. Which of the 5 pictures cannot be right unless the flag is torn?
THURSDAY, FEBRUARY 5, 2009 Week 1 - Problem 2
The face of a clock is cracked into 4 pieces. The sums within the parts are consecutive numbers. Provided there is only one possible way to crack it, the face would look like:
WEDNESDAY, FEBRUARY 4, 2009 Week 1 - Problem 1
ABCD is a square. Its side is equal to 10cm. AMTD is a rectangle. Its shorter side is equal to 3 cm. How many centimetres is the perimeter of the square ABCD larger than that of the rectangle AMTD? A.14 cm. B. 10 cm. C. 7 cm. D. 6 cm. E. 4 cm
http://kangaroo.math.ca/samples/workingbackward/index.html Question 1 Marissa wrote her favorite number in the dark cloud and performed correctly several calculations following the sequence in the diagram. What is Marissa’s favorite number?
Question 2 The three members of a rabbit family have altogether eaten some number of carrots for breakfast. Father woke up first and ate half of the carrots. Mother woke up next and ate seven of the remaining carrots. Son Bunny woke up last and ate the remaining four carrots. How many carrots had the family eaten? First off, here is a chart of the carrot eating that was going on:
An example problem There were several birds on two trees. First, five birds flew from the first tree to the second tree. Second, one bird flew away from the second tree. Third, four birds from the second tree flew to the first one. In the end, there were six birds on each tree. How many birds were on each tree at first? The forward process The first step in solving "backward" problems is converting the question which is usually given in words into a diagram of symbols that will help you easily see what steps you should take to get to the solution. Here is the arrow diagram of what was taking place in this problem:
We see that, after the first step, the number of birds on the first tree decreased by 5, while the number of birds on the second tree increased by 5. After the second step, the number of birds on the second tree decreased by 1. On the third step, the number of birds on the second tree decreased by 4, while the number of birds on the first tree increased by 4 birds from the second tree that moved there (this step is illustrated on the diagram by the slanted arrow. At the end, there are 6 birds on each tree. Progressing backward So we had drawn a symbol's chart and placed the problem operationsin it to produce this diagram:
We now have to find the inverse operations for these operations and we will put them in a diagram too. So for example where we have a "+4" arrow going into the top-right tree, we will find the inverse (or opposite) which becomes a "-4" arrow going away from the
top-right tree. Applying similar ideas, we then get the following diagram below:
Everything together Now, let us reverse the steps, one at a time, by doing the opposite (following the thick arrows):
If we take the top tree for example, starting at "6" at the top-right corner, we follow the backward arrows and perform: 6 - 4 = 2, 2 + 5 =7. For the bottom tree, again starting from the bottom-right tree and following the backward arrows, we will get: 6 + 4 = 10, 10 + 1 = 11, 11 - 5 = 6. Thus there were 7 birds on the first tree and 6 birds on the second tree at the start!!!
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