MSP MATH WIZARD 2011 : Elimination Round
February 26, 2011 On the answer sheet provided, write your answer in chronological chronological order. Time allotted: 1.5 hours 1. How How man many y min minut utes es is it unti untill 5:0 5:00 0 if if for forty ty minu minute tes s ago ago it was was fou fourr tim times es as many many minutes past 2 o’clock ? 2. A set set of cons consec ecut utiv ive e pos posit itiv ive e int integ eger ers s beg begin inni ning ng with with 1 iis s wri writt tten en on on the the teac teache her’ r’s s blackboard. A student came along and erased one number. The average of the remaining numbers is 35 and 7/17. What was erased? 3. For ho how many va values of b, b, 0 < b < 24 24, co could p 2 number p?
≡
b (mod 24) for some prime
4. Jack Jack and and Jil Jilll wen wentt up up the the hill hill at 4mph 4mph.. The They y sta start rted ed tumb tumbli ling ng down down at 6mph 6mph.. Unfortunately, Unfortunately, they hit a rock (and broke their crowns) at exactly halfway down the hill. What was their average speed in mph during the trip up and halfway down? 5. A number N, expressed in base (A + 1) is AAAA. If N = Q(Q – 2), what is Q expressed in base (A +1)?
6. The 10th term of an arithmetic sequence is 27. The 20 th term is 12. Find the 40 th term.
7. A coin is biased so so that a head head is three three times as likely to occur occur as a tail. Find the expected number of tails when this coin is tossed twice. 8. Evaluate
lim
x
−e
x →0
xe
x
x
+1
− x
9. Suppose that that 25% of all the wise wise people are nice and and half of all the nice nice people are are wise. Suppose further that 25% of all the people are neither wise nor nice. What percent of all the people are both wise and nice?
10. An equilateral triangle has sides of length 4 units. Another equilateral triangle is constructed by drawing line segments through the midpoints of the sides of the first triangle. If this procedure can be repeated an infinite number of times, what is the total perimeter of all the triangles formed?
11.Dr. Rex Calingasan likes to eat chocolates for an afternoon snack. On Monday morning he brings in a bag of 5 chocolates, 3 with red wrappers and 2 with green wrappers. At snack time every day, he reaches into the bag, pulls one out and eats it. What is the probability that the chocolate he eats on Friday will have a red wrapper?
12.At 1:00 pm on February 25, 2011, a dog fall asleep. It sleeps for 5 hours, then is awake for 5 hours, then sleeps for 5 hours, etc. At 1:17 pm on the same day, a cat falls asleep. It sleeps for 2 hours, then is awake for 2 hours, then sleeps for 2 hours, etc. Determine the largest integer H so that in every unbroken 24 hour period in the year 2011 there must be a total of at least H hours during which both animals will be asleep. 13.Find the minimum length of the line that can be drawn tangent to the ellipse 9x 2 + 4y2 = 36 at the first quadrant. 14.In a class of 100 students, there are 50 who play basketball, 45 who play billiard and 50 who play tennis. Only 15 of these students play all 3 sports. Everyone plays at least one of these sports. How many of the students play exactly two of these sports?
15.What is the area of the circle whose equation is 16x 2 + 16y2 - 16x + 8y – 59 = 0?
2 1 1 −1 16.Consider the range of the inverse sine function as − π , π , find sin cos π . 3 2 2
17.Suppose that 10 teams participated in a soccer tournament where each team played exactly one game with each of the other teams. The winner of each game received 3 points, while the loser received 0 points. In case a tie, both teams received 1 point. At the end of the tournament, the 10 teams received a total of 130 points. How many games ended in a tie? 18.The lines AB and CD are parallel and a distance 4 apart. Suppose that AD intersects BC at a point P between the lines. IF AB = 4 and CD = 12, how far is P from the line CD? 19.Let a and b be real numbers with a > 1 and b >1. Let log a(b) denote the logarithm of b for the base a. Four students compute the log a (b).logb (a). The first gets the answer 1. The second gets logab (ab). The third get a b ba . The fourth gets log b (a) + loga (b). How many of the students are correct? 20.A bee woke up on a Sunday morning and went directly to work. It flew straight south for 1 hour to a nice sweet field and spent 30 minutes there. Then it went directly west for ¾ hour to a garden where it stayed for 1 hour. After it flew the shortest path home. Assuming that the bee flew with constant speed and that the earth is flat, for how long was the bee way from home?
God bless!!! MSP-NCR Math Wizard 2011 Committee
