Đề thi FMO vòng chung kết quốc tế 2022 Khối 9 (Kèm đáp án)

 FMO – GRADE 9 
 I. Warm-up (3 points per question / No points deducted for wrong answers) 
1. Solve the puzzle below to find the value of FMO. 
 A) 16 B) 17 C) 18 D) 19 E) 20 
2. Danny wants to draw triangles whose 3 sides are different 1-digit prime numbers. How 
 many different triangles can he draw? 
 A) 0 B) 1 C) 2 D) 3 E) 4 
3. The pie diagram below shows how a baby spends her time in one day. Find the percent 
 of time she spends sleeping in a day. (Round to the nearest integer) 
 A) 43% B) 44% C) 45% D) 46% E) 47% 
4. The length of a rectangle is 5 times as the width. If the length is decreased by 20cm 
 and the width is increased by 20cm, the area will increase by 1200 cm2. Find the original 
 area of the rectangle. 
 A) 720 B) 1000 C) 500 D) 1125 E) 2000 
5. Given a function ( ) = 2 − + 4. If (1) = 5 and (−1) = 7, calculate (2). 
 A) 10 B) 12 C) 8 D) 4 E) 14 
6. How many positive integers N satisfy that exactly one of the numbers N and (N + 30) is 
 a 3-digit number? 
 A) 29 B) 30 C) 60 D) 58 E) 62 
 1 
 FERMAT Mathematical Olympiad Summer 2022 
 All materials on the test are copyrighted and owned by Fermat Education 
 II. Speed-up (4 points per question / No points deducted for wrong answer 
11. In the diagram below, the square OMNP has side length 6cm. The line with equation 
 1
 yx intersects MN at A. The line with equation yx2 intersects NP at B. Find the 
 3
 area of the shaded region. 
 A) 24 B) 30 C) 18 D) 27 E) 21 
12. Fermat wrote down three positive numbers. He chose one of them and added it to the 
 average of the other two. He repeated this for each of the three numbers and got the 
 results 42, 44 and 48. What was the largest number written by Fermat? 
 A) 29 B) 21 C) 17 D) 31 E) 27 
13. For numbers a and b , the notation ab means a22 ab b . For example, we have 
 the operation 1 2 122 1 2 2 7 . If (x 2) x ( x 2)2 , how many different 
 values of x are there? 
 A) 0 B) 2 C) 1 D) 3 E) 4 
14. A square is inscribed in a semi-circle with diameter 1 as the figure below. What is the 
 area of the square? 
 2 1 1 1 1
 A) B) C) D) E) 
 5 20 5 3 12
15. For a positive integer n , fn() is defined as the exponent of the largest power of 2 that 
 divides n . For example, f (40) 3 as 20 23 5. What is the value of fN() given that 
 N 1 2 3 ... 2022 2023? 
 A) 2023 B) 1011 C) 2014 D) 1516 E) 2013 
 3 
 FERMAT Mathematical Olympiad Summer 2022 
 All materials on the test are copyrighted and owned by Fermat Education 
IV. Star of hope (10 points per question/ 10 points deducted for wrong answer) 
 In this part, you should write your answer on the answer sheet. 
 21. A group of 50 nurses had to do at least one of three shifts including the morning 
 shift, the afternoon shift and the night shift. The results are listed below: 
 The number of nurses 
 The morning shift 37 
 The afternoon shift 42 
 The night shift 30 
 At least how many nurses did all 3 shifts in that day? 
 Congratulations! 
 This is the end of the test. Please review the answers carefully before submission. 
 5 
 FERMAT Mathematical Olympiad Summer 2022 
 All materials on the test are copyrighted and owned by Fermat Education