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Singapore Mathematical Society Singapore Matheniatical Olympiad (SMO) 2OL5 Senior Section (Round 1) Wednesday, 3 June 2015 0930 - 1200 hrs Instructions to contestants 1. Answer ALL 35 quest'ions. 2. Enter your o,nswers on th,e answqr sheet proui'd,ed. 3. For the multi,ple cho'ice quest'i,ons, enter your answer on the answeT' sheet by shad,i,ng the bubble conta'ini,ng the letter (A, B, C, D or E) correspbndi'y't'g to the correct answer. /t. For the other short questions, write Aour o,nswer on the answer sheet and shade the appropriate bubble below your ansuer. 5. No steps are needed to justi,fy Aour answers. 6. Each quest'ion carries 1 rnark. 7. No calculators are allowed. 8. Throughout this paper, let lr) denote the greatest i,nteger less than or equal to r. For erample, 12.1): 2, 13.91 : 3. PLEASE DO NOT TURN OVER LTNTIL YOU ARE TOLD TO DO SO.

- \ Multiple Choice Questions' .,ffi+T Jbo-7 1. Find the exact value of 6- - ^ *, (A) 1e7 (B) 1e8 (c) 1ee (D) 200 (E) 201 2 Simprify ##- @)* "_lYjq (c) * Dry @)T 3. Suppose m and. n are real numbers such that the roots of the equation 2r2 -mr*8 : 0 are a and B while the roots of the equation 5r2 - LLr *5n : 0 ur" 1-urra l. fina a.p the value of mn- 1 (A) ; (B) 4 (c) 8 (D) 12 (E) 16 4. Find the largest number among the following numbers: (A) /8 + /8 @) rt + JO (c) ./6 + /10 (D) /5 + '/i (E) '/d + '/n 5. Which of the following is true? (A) cos 221" > sin319" (B) cos 139" > sin221" (C) sin139' > cos41o (D) sin 22I" > cos 139' (E) sin41" > cos 319o 6. Find the smallest number among the following numbers: (A) log2e15 2016 (B) log2s16 2017 (C) Io92617 2018 (D) log2e16 2019 (E) log261e 2020 7 . If n and y are nbn-zero numbers such that r ) A, which of the following is always true? 11 t^);.y (B) I>r (c) l"l >lsl @)#,h @);rI s. If f.(r):2112- 1l - r-r,where 03r" l, nna the range of f (r)' - -z', (A) -3S/(r) sr (B)-3

F t 9. How many prime numbers.p satisfy the inequality lz+togrToltr' 3, (A) 6 (B) 7 (c) 8 (D) e (E) 10 10. When a polynomiat f (") is divided by (" - 1) and (r * 5), the remainders are -6 and 6 respectively. Let r(r) be the remainder when /(z) is divided by ,2 * 4r - 5. Find the value of r(-2). (A) 0 (B) 1 (c) 2 (D) 3 (E) 5 Short Questions - 11. Find the value of 2*1og23,3*1og34 1+f,"gr3-1+lrrg,2 1 12. Find the absolute vaiue of the coefficierrt of I in the expansion of T z ,r 1o t - (2".'- !\ rl 13. If a * b :4 "oa (f + r/-a)(r + \/b) : +,find the vatue of ab. 4\v/\'2 14. Find the value of.pif there is a unique value of r satisfying the equation'p2r+r * 1: 1/Q4salra'P. 15. Suppose r and y arc real numbers such that 12 and y2 arepositive integers. Find the maximum value of 12 - ry if. 16. Find the smaliest positive integer z (measured in degrees) such that tan(r- 160") : #ffi; 17. Frnd the smallest positive integer k such that 111 t"g# r01il * r"** zorsr +" " + l.ogml# ,015t > 2015'

18. Let a

Consider the foliowing equations: i1_-l__ It az 11_r_ lj 11__l__o I o'yo 1 a 1tv',1 -n 28. Find the sum of (r+I)a(g * 1)a over all possible ordered triples (r,a,") that satisfy the above three equations simultaneously. The diagram below shows three right-angled triangles, where BC : 14, GF - 10, DE :7 and IBCA: IBDE : IFGD : d. Find the madmum possible value of AB+BD+DF. The diagram below shows a rectangle ABCD such that E is the midpoint of. BC and -F is the midpoint of. CD. The diagonal BD intersects AF and AE at e and 7 respectively. The vertical iine PS passing through Q is perpendicular to AB and, intersects AE at.R. It is also given that AB: CD:12 cm and BC: AD:6 cm. Find the area of the triangle AQRT in cm2. 29.

- "\ Find the minimum value of 13secd - gsin 0tan0 fo, -1 < 0 < 2 Irr the figure below, the line BD is tangent to the circle at C. through the centre O of the circle and intersects the circle at ICDE: 34" and IDCE: ro. Find the value of r. ,, 31. The line ,4D passes E. It is given that 32. In the figure below, A, B , C and D are points on the circle such that the straight lines AB and CD intersect at E. Let IBCEI arrdIADEI denote the areas of the triangles ABCE and. AADE resoectivelv. y E!o' Vft: 25 and AE :1 crn' find the length of the line CE in cm.

33. 34. In how many ways carr a group of B dift'ererrt guests (corrsistirrg of,4 males and 4 females) be seated at a round tabie with B seats such that there are exactly 3 males who are seated next to each other? Find the number of rectangies that can be formed from the gridlines of the board as shown irr the figure below. tr.itj. Let X = {1,2,3,4,5,6,7}, and let A: {Ft,Fz,...,F,-} be a collection of distinct subsets of X such'that the intersection 4a Fi contains exactly one element whenever i, + j. For each i, e X,let r; be the number of elements in ,4 which contains 'd. Suppose rI : 12 : l, TB - 14 : rS : 16: 2 and T'T : 4. Find the value of n2\- n.