Đề thi FMO Senior Secondary
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FMO – Senior Secondary I. Warm-up (3 points per question / No points deducted for wrong answers) 1. The sum of seven digits of the number xxxyyyy is equal to the two-digit number xy. What is the sum of x and y ? A) 10 B) 7 C) 13 D) 5 E) 12 2. Teacher gives his students one test every week. After the first five tests, Fermat had an average test score of 14. On each of the next n tests, his score was 20. After all these tests, his average test score was 18. How many tests did Fermat take in total? A) 10 B) 16 C) 15 D) 18 E) 35 3. A store sells pencils and ball pens. Before a price increase, the price of each pencil is $1.50 and the price of each ball pen is $2.00. The price of each pencil is increased by 10% and the price of each ball pen is increased by 5%. What is the percentage increase in the total cost of a purchase of 2 pencils and 1 ball pen? A) 10% B) 12% C) 4% D) 8% E) 25% 4. 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 , find the maximum value of x . A) 0 B) 2 C) 7 D) 3 E) 5 5. Given a cyclic quadrilateral ABCD with AB AC and CB CD . If ABC 32o , find the measure of BCD. A) 52o B) 64o C) 54o D) 62o E) 58o 6. The two roots of the equation x2 33 x m 0 are both prime numbers. Given that m is a natural number, find the sum of digits of m ? A) 14 B) 8 C) 5 D) 62 E) 6 1 FERMAT Mathematical Olympiad Summer 2022 All materials on the test are copyrighted and owned by Fermat Education 13. Trapezoid ABCD has vertices ABCD(0,0); (12,0); (11,5); (2,5). The sides AD and BC are extended to intersect at the point E . Find the area of triangle EAB . A) 240 B) 180 C) 150 D) 144 E) 120 14. Twelve identical semicircles are drawn inside a square of side length 12cm as below. Find the area of the shaded region. A) 8 16 B) 24 16 C) 16 8 D) 16 16 E) 8 32 15. A palindrome is a positive integer that is the same when read forwards or backwards. For example, 1441 and 252 are palindromes. How many 4-digit palindromes greater than 2023 are there? A) 80 B) 79 C) 72 D) 71 E) 69 III. Challenge (8 points per question / No points deducted for wrong answers) 16. A function f is defined as below: • If n is an odd integer then f( n ) 2 n 1. • If n is an even integer then f( n ) n2 1. For example, f (1) 2 1 1 3 and f ( 2) ( 2)2 1 3. How many integers n satisfy that f( f ( n )) 7 ? A) 0 B) 1 C) 2 D) 3 E) 4 17. Given 3 identical cookies perfectly fit inside a rectangular tray as below. If the diameter of each cookie is 4cm, find the area of the tray. A) 16 16 3 B) 16 32 3 C) 16 8 3 D) 32 16 3 E) 32 8 3 3 FERMAT Mathematical Olympiad Summer 2022 All materials on the test are copyrighted and owned by Fermat Education
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