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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2022 (Open Section, Round 1) Thursda y, 2 June 2022 0930-1200 hrs Instru c ti ons to contestants 1. Answer ALL 25 questi o ns. 2 . Write your a nswers in the a ns w er sheet prov i ded and shade the approp ri ate bubble s below your answers. 3. No steps are nee ded to j ust ify yo u r answers . 4. Each questio n carries 1 ma rk . 5 . No calculators a re a llo we d . PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO Co-organizer Department of Mathematics, NUS Sponsored by Micron 1

In this paper, let 股 denote the set of all real numbers, and 杠」 denote the greatest integer not exceeding x. For examples, LS」= s, L2.8 」= 2, and L-2.3 」=-3. 1. If S = •~~021 10• l+ l'find l 2S 」 2. All the positive integers 1, 2, 3, 4, · · ·, are grouped in the following way: G1 = {1, 2}, G2 = {3 , 4 , 5, 6}, G3 = {7, 8, 9, 10, 11, 12, 13, 14}, that is, the set Gn contains the next 2n positive integers listed in 郘 cending order after the set G正 1, n > 1. If S is s the sum of all the positive integers from G1 to G8 , find l司· 3. A sequence of one hundred positive integers x1, x2, x3, • • •, x100 are such that (x1)2 + (2x 沪+ (3x 扩+ (4x 矿+· · ·+ (lOOx100)2 = 338350. Find the largest possible value of X1 + x2 + X3 +· · ·+ X100- 4 . Let a and b be two real numbers satisfying a < b, and such that for each real number 2 m satisfying a < m < b, the circle x + (y -m)2 = 25 meets the parabola 4y = x2 at four distinct points in the Cartesian plane. Let S be the maximum possible value of b - a. Find l4S 」. 5 . L et P be a point within a rectangle ABCD such that PA = 10, PB = 14 and PD = 5 , 邸 shown below . Find lPCJ. A D 14 B c 6. In th e diagram below, the rectangle ABCD has area 180 and both triangles ABE an d AD F have are 邸 60. Find the area of triangle AEF. A B 、、、、｀ D F c E 2

7. A tetrahedron in IR3 has one vertex at the origin O and the other vertices at the points A(6, 0 , 0) , B(4, 2 , 4) and C(3, 2, 6). If xis the height of the tetrahedron from 0 to the plane ABC, find l 5x 勹 8 . L et x and y be real numbers such that (x - 2)2 + (y - 3)2 = 4. If S is the largest p o ss ible value of x2 + y2 , find l (S - 17) 勹 9. Let S be the maximum value of w3 -3w subject to the condition that w扛 9'.S 10w2. Find LS 」. 10 . In the quadrilateral ABCD below , it is given that AB = BC= CD and 乙ABC = 80°a nd 乙B C D = 160 °. Suppo se 乙ADC= x0. Find the value of x. B A D 11. Let a, b, c be in t egers with ab+ c = 49 and a + be = 50 . Find the largest possible value of ab c. 12. Find t h e l arges t possible value of i al + lbl , where a and bare copr i me integers (i.e., a and b are integ e rs which have no common factors larger than 1) such that i is a solution of t he equation below : J 4x+ 5— 4石言＋ ✓ x+ 2- 2石言= 1. 13. Let S be t he s et of re al solu tion s (x, y , z ) of the following s y stem of equations: 4x 2 1 + 4x2 =y , 4y2 1 + 4y2 = z, 4 z 2 =x . 1 + 4z2 For each (x, y , z) E 5 , de fine m( x , y , z ) = 2000 (lxl + I Y I + lzl). De termine th e maximum value of m(x, y, z) ov er all ( x, y , z ) E 5 3

14. Assume that tis a positive solution to the equation t~ ✓ l+ ✓ l+~ Determine the value of t4 - t3 - t + 10. 15 . In the triangle ABC shown in the diagram below , the external angle bisectors of 乙Band 乙C meet at the point D. The tangent from D to the incircle w of the triangle ABC touches w at E, where E and Bare on the same side of the line AD. Suppose 乙BEG= 112°. Find the size of LA in degrees. w A C D 16 . Find the largest integer n such u止LL n " + 5n - 9486 = 10s(n), where s(n) is the product of all digits of n in the decimal representation of n. (For examp le, s(481) = 4 x 8 x 1 = 32 .) 17. Find the number of integer solutions to the equation 19x + 93y = 4xy. 18 . Find the number of integer solutions to the equation x1 + x2 - x3 = 20 with x1~ X2~X3~0. 4

／了 19 . In the diagram below, Eis a poin t outside a square ABCD such that CE is parallel to BD , BE = BD , and BE intersects CD at H. Given BE= 看＋迈， find the length of DH. A D E l. 20 . The diagram below shows the region R = {(x, y) E 胶2I Y~ ½ 沪} on the xy-plane bounded by the parabola y =½x 2. Let C1 be the largest circle lying inside R with its lowest point at the origin. Let C2 be the largest circle lying inside R and resting on top of C1. Find the sum of radii of C1 and C2. 0 1. Find the smallest positive integer x such that 3x2 + x = 4沪+ y for some positive integer y. 22 . A group of students part 沁 pate in some sports activities among 6 different types of sports. It is known that for each sports activity there are exactly 100 students in the group participating in it; and the union of all the sports activities participated by any two students is NOT the entire set of 6 sports activities. Determine the minimum number of students in the group. 23 . Let p and q be positive prime integers such that p3 -5忙- l8p = q9 -7 q. D ete rmine the smallest value of p. 5

、七 24 . Given that a, b, care positive real numbers such that a+b+c = 9, find the maximum value of a2b3 伎 25. Let JR+ be the set of all positive real numbers . Let f : 胶十分 JR+ be a function satisfying xyf(x ) (f(y) - f(yf(x))) = 1 1 for all x,y E 厌七 Find f( —一2022 ) . 6