Đề vòng chung kết Quốc gia kỳ thi Olympic toán học Quốc tế TIMO 2020-2021 Khối 8 (Kèm đáp án)

 THAILAND INTERNATIONAL 
 MATHEMATICAL OLYMPIAD 
 FINAL ROUND 2020 – 2021 
 Secondary 3 
 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. 
 All answers should be written on the ANSWER SHEET. 
 Number Theory 
13. It is known that x is rational, x 0 and x 9999 . Find the value of x. 
14. If a 4-digit number has exactly 3 positive factors, find its maximum value. 
15. If k 7 268,435,456, find the value of k. 
16. Now is Saturday. Which day of the week will it be after 552021 days? 
17. Find the remainder when 34812021 is divided by 60. 
 1 1 1
18. Find the number of sets of positive integral solution(s) of . 
 ab2021
 Geometry 
19. Find the area of an equilateral triangle with perimeter 18. (Answer in simplest surd form) 
20. Find the area surrounded by the line with equation xy 2021 2021 20 in the 
 coordinate plane. 
21. G is the circumcenter of ABC , where AB 12, AC 35, BC 37 . Find the length of AG. 
22. Find the number(s) of points with integral coordinates in the coordinate plane that lie on 
 the circle with radius 10, and with center at origin. 
23. I is the incentre of ΔPQR. If  QIR 105 , find QPR . 
24. Find the radius of a circle inscribed in a rhombus with diagonals of length 14 and 48. 
 Combinatorics 
25. A dice is thrown 4 times. Find the probability that the numbers obtained in each time are 
 the same. 
26. Suppose 2 cards are drawn from an ordinary poker deck of 52 playing cards one by one 
 (without replacement). Find the probability of getting the same number or letter. 
27. Find the number of 3-digit positive numbers that have product of digits equals to 36. 
28. Find the number(s) of integer x such that xx2 10 21. 
29. Find the number of ways to arrange 4 identical bowls and 4 identical cups in a row. 
30. Let xn be a positive integer for all n. How many solution sets are there in the inequality 
 x1 x 2 x 3 x 4 19 ? 
 ~ End of Paper ~