Đề vòng chung kết Quốc gia kỳ thi Olympic toán học Quốc tế TIMO 2020-2021 Khối 9 (Kèm đáp án)

 THAILAND INTERNATIONAL 
 MATHEMATICAL OLYMPIAD 
 FINAL ROUND 2020 – 2021 
 Senior Secondary 
 Time allowed: 120 minutes 
 Question Paper 
 Instructions to Contestants: 
1. Each contestant should have ONE Question-Answer Book which CANNOT be 
 taken away. 
2. There are 5 exam areas and 6 questions in each exam area. There are a total of 30 
 questions in this Question-Answer Book. Each question carries 5 marks. Total 
 score is 150 marks. No points are deducted for incorrect answers. 
3. All answers should be written on ANSWER SHEET. 
4. NO calculators can be used during the contest. 
5. All figures in the paper are not necessarily drawn to scale. 
6. Write down the answer in the simplest form. If the calculation result is a fraction, 
 please write down the answer as a proper or mixed fraction, decimal figure is also 
 accepted. Marks will NOT be given for incorrect unit. 
7. This Question-Answer Book will be collected at the end of the contest. 
 DO NOT turn over this Question-Answer Book without approval of the examiner. 
 Otherwise, contestant may be DISQUALIFIED. 
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 All answers should be written on the ANSWER SHEET. 
16. Find the numbers of positive simplified fractions with denominator 2021. 
17. It is known that n! n ( n 1) ( n 2) ... 1. Find the number of “0” at the end of the 
 factorial 2021!. 
18. If k9 68,719,476,736, find the value of k. 
 Geometry 
19. I is the incentre of ΔPQR. If  QIR 110 , find QPR . 
20. G is the circumcenter of triangle ABC , where AB 8 , AC 15, BC 17 . Find the 
 length of AG. 
21. Find the area surrounded by the line with equation xy 2021 2021 40 in the 
 coordinate plane. 
22. Find the radius of a circle inscribed in a rhombus with diagonals of length 24 and 32. 
23. Find the number(s) of points with integral coordinates in the coordinate plane that lie on 
 the circle with radius 13, and with center at the origin. 
24. Let lengths of three sides of an obtuse triangle be 12, 14 and x, where x is an integer. Find 
 the number of possible values of x. 
 Combinatorics 
25. Mrs. Tsang has 4 children. Find the probability that exactly one of the children is a boy. 
26. Find the number of the permutation(s) when you arrange 5 identical bowls and 5 identical 
 cups in a row. 
27. Find the numbers of set(s) of positive integral solutions such that a b c d 20 . 
 1 1 1
28. Find the number of sets of integral solution(s) of . 
 ab2021
29. Find the number of ways to assign 14 people into seven groups evenly. 
30. Suppose 2 cards are drawn from an ordinary poker deck of 52 playing cards one by one 
 (with replacement). Find the probability of getting the same number or letter. 
 ~ End of Paper ~ 
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