SASMO 2013 Sec 3 Contest

Division

Mathematical Olympiads

S3 Name:

2013 _________________________________

(as in Birth Cert)

NRIC:

______________

Class:

______________

Index Number: ________

School: __________________________________

INSTRUCTIONS 1. Please DO NOT OPEN the contest booklet until the Proctor has given permission to start. 2. TIME : 1 hour 30 minutes

3. Attempt all 25 questions. Show your working clearly.

4. Write your answers neatly in the answers sheet in the booklet. 5. PROCTORING : No one may help any student in any way during the contest. 6. Only 2B pencil may be used. No other materials, including calculators, are allowed. 7. All students must fill in your Name, NRIC no, Index Number, Class and School.

8. MINIMUM TIME: Students must stay in the exam hall at least 1h 15 min.

SASMO 2013, Secondary 3 Contest 9. No exam papers and written notes can be taken out by any contestant. 10. All correct answers score 1 point each and are no marks deducted for incorrect answers.

EACH QUESTION Scores 1 point each 1. Tom must travel from A to B and he plans to go at a certain speed. He would like to arrive earlier than planned and notes that traveling at a speed 5 km/h faster than planned, he will arrive 5 hours earlier and traveling at a speed 10 km/h faster than planned, he will arrive 8 hours earlier. What is his planned speed?

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2. On the island of nobles and liars, 50 people are standing in a queue. Everyone, except the first person in the queue, said, that the person before him in the queue is a liar. The first man in the queue, said, that all people, standing after him are liars. How many liars are in the queue? (Nobles always speak the truth, and liars always tell lies.)

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1

SASMO 2013, Secondary 3 Contest 3. In a football league with 6 teams (A, B, C, D, E, F), each team plays each other team exactly once. So far, A has played one match, B has played 2 matches, C has played 3 matches, D has played 4 matches, and E has played 5 matches. How many matches has F played so far?

4. For how many integers ?

does there exists a convex -gon, whose angles are in ratio

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5. A sequence of numbers has 6 as its first term, and every term after the first is defined as follows: If a term, A, is even, the next term in the sequence is . If a term, B, is odd, the next term is . Thus, the first four terms in the sequence are 6, 3, 10, 5. What is the th 200 term of the sequence?

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2

SASMO 2013, Secondary 3 Contest

6. Tom rolls a 6-sided die. Jerry rolls a second 6-sided die. Tom wins if the values shown differ by 1. What is the probability that Tom wins? (Express your answer in fractions)

7. For how many integers

, such that

is a perfect square?

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8. How many positive integers not exceeding 2013 are multiples of 3 and 4 but not 5?

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9. How many right-angled triangles can be formed by joining three vertices of a given regular 14-gon?

3


10. The first element of the sequence the value of if .

, and if

then

. Find

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11. How many 10-digit numbers only composed of 1, 2, and 3 exist, in which any two neighbouring digits differ by 1?

4


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12. Four different numbers , , , and are chosen from the list What is the largest possible value for the expression

and

13. Evaluate the following sum

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, find

.

5


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15. Let be the least number with the following property: is a perfect square and a perfect cube. How many positive factors does the number have?

is

16. Two circles have their centers on the same diagonal of a square. They touch each other and the sides of the squares as shown. The square has a side of length 1 cm. What is the sum of the lengths of the radii of the circles?

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17. What is the last digit of

6


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18. How many 7-digit numbers contain at least one 7?

19. Evaluate (in the simplest form)

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20. How many zeros does the following product

end with? 7


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21. In isosceles trapezoid ( is parallel to ), is the midpoint of side and angle . Find the perimeter of the trapezoid.

22. A palindrome is a positive integer that is the same when read forwards or backwards. For example, three palindromes are 8; 343 and 5578755. Determine the number of palindromes less than 10 000?

8


______________________________________________________________________________ 23. The vertices of triangle are , that . Find the value of

, and .

. Points

24. Find positive solutions of the following equation .

9

and

are on

so


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25. A sequence of numbers

satisfies 1. 2.

Find the value of

.

End of Paper

Name: _________________________ Class: _________________

10


1. 14. 2. 15. 3. 16. 4. 17. 5. 18. 6. 19. 7. 20. 8. 21. 9. 22. 10. 23. 11. 24. 12. 25. 13.

Rough Working SASMO 2013, Secondary 3 Answers 11


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

15 km/h 25 3 matches 2 2

103 134

84 4027 64

13.

√

14. 15. 16.

4x6x3=72

17. 18. 19. 20. 21. 22.

√ √ 1

√

61 249 6 198

12


23. 24.

65 √

√

25.

13

SASMO 2013 Sec 3 Contest

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