Download [PDF] 2022 SMO Senior Section Round 1 Questions Singapore Mathematical Olympiad

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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2022 Senior Section (Round 1) Wednesday, 1 June 2022 0930 - 1200 hrs Instructions to contestants 1. Answer ALL 25 questions . 2 . Enter your answers on the answ er sheet provided. 3 . For the multiple choice questions , enter your answer on the answer sheet by shading t h e bubble containing the letter (A, B, C, D or E) corresponding to the co飞 ct answer . 4. For the other short questions, write your answer on the answer sheet and shade the appropriate bubble below your answer . 5 . No steps are needed to justify your answers. 6 . Each question carries 1 mark. 7 . No . calculators are allowed. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO. Co -organizer Department of Mathematics, NUS Sponsored by Micron 1

Multiple Choice Questions X 2 m n 1. Suppose the roots of —+ mx + n = 0 are - and- . Find the smallest value of mn. 2 2 3 (A ) - 1080 (B) -90 (C) 0 (D) 90 (E) 1080 2 . Which of the following is true? (A)归;

9. Suppose tan2 x - tanx +尽 y= tan 五+ tanx + 1 ' where —90 ° < x < 90°. Find the maximum possible value of J: 丽(y -5). 10 . In the figure below, PQRS is a square inscribed in a circle. Let W be a point on the arc PQ such that W S =迈 .Find (WP+ WR)2. w 11. The figure below shows a quadrilateral ABCD such that AC= BD and P and Qare the midpoints of the sides AD and BC respectively. The lines PQ and AC meet at R and the lines BD and AC meet at S. If 乙PRC = 130°, find the angle 乙DSC (in°). D A B 12 . How many distinct terms are there if (丑＋沪） ll(xll + yll 沪 is algebraically expanded and simplified? 13. If v'x2 + 7x -4 + v'x2 -x + 4 = x - 1, find the value of 3x 耳 14x. 14 . Let k = -1十三， and let f(x) = (k2 + 2k 1+ 2)10 工 Find the value of log2022 f (2022). 15. Find the smallest odd integer N, where N > 2022, such that when 1808, 2022 and N are each divided by a positive integer p, where p > 1, they all leave the same remainder. 3

12 48 16 . If - + - = 1, where x and y are positive real numbers, find the smallest possible X y value of x + y. 17 . Find the largest value of 40x + 60y if x - y S 2, 5x + y 2='. 5 and 5x + 3y S 15 . 18 . Suppose l COS X - cosy = - , 2 1 3 sinx - siny = m m If sin(x + y) = -, where - is expressed as a fraction in its lowest terms, find the n n value of m + n. 19 . For some positive integer n, the number n3 -3n 红 3n has a units digit of 6. Find the product of the last two digits of the number 7(n - 1)12 + l. 20. Find th 1 20n + 2020 e argest positive mteger n for which 3n —6 is a positive integer. 21. In the xy-coordinate system, there are two circles passing through the point (11 , 3戎）， and each of these circles is tangent to both the x-axis and the line y =戎 x. Let S be the sum of the radii of the two circles. Find 戎 s. 22 . L et P and Q be the points (20(v5 -1), 0) and (0, 10( 石- 1)) on the xy-plane. Let R be the point (a , b ). If 乙PRQ is a right angle, find the maximum possible value of b. 23 . How many positive integers n do not satisfy the inequality 社 log20n >石？ 24 . Let f(x) be a funct ion such that 3f( 沪） + /(13 -4x) = 3x2 - 4x + 293 for all re al number x. Find the value off (1) . 25 . Fin d the largest positive integer M such that 2·2 M cos x - sm x + sin x = - 888 h邸 areal solution . 4