(as in Birth Cert)
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. questions. Show your your working working clearly. 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, Name, NRIC no, Index Index Number, Number, Class and 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 10. All correct correct answers answers score 1 point each and and are no marks marks deducted deducted for incorrect 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 standing after him are liars. How many liars are 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 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 the next term in in the sequence 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) 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
SASMO 2013, Secondary 3 Contest
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
SASMO 2013, Secondary 3 Contest
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12. Four different numbers numbers , , , and are chosen 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
SASMO 2013, Secondary 3 Contest
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15. Let be the least number with the following property: is a perfect square and a perfect perfect cube. How many positive factors does the number 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 l ength 1 cm. What is the sum of the lengths l engths of the radii of the circles?
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17. What is the last digit of
6
SASMO 2013, Secondary 3 Contest
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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
SASMO 2013, Secondary 3 Contest
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21. In isos isosce cele less trap trapez ezoi oid d ( is para paralle llell to ), is the midpoint of side and angle . Find the perimeter of the trapezoid.
22. A palindrome is a positive integer that is i s 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?
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SASMO 2013, Secondary 3 Contest
______________________________________________________________________________ 23. The vertices of triangle that
are
,
, and
. Find the value of
. Points
.
24. Find positive solutions of the following equation .
9
and
are on
so
SASMO 2013, Secondary 3 Contest
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25. A sequence of numbers
satisfies 1. 2.
Find the value of
.
End of Paper
Name: _________________________ Class: _________________ ___________ ______
10
SASMO 2013, Secondary 3 Contest
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
SASMO 2013, Secondary 3 Contest
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. 17. 18. 19. 20. 21. 22.
4x6x3=72
√ √
√
1
61 249 6 198
12
SASMO 2013, Secondary 3 Contest
23. 24.
65
√
√
25.
13
