[PDF] Open Section - First Round - SMO Singapore Mathematical Olympiad 2014.pdf | Plain Text

Singapore Mathematical SocietY Singapore Mathematical Olyrnpiad (SIr{O) 2014 (Open Section, First round) 'Wednesday, 4 June 2014 0930-1200 hrs Instructions to contestants 1, Answer ALL 25 questions. 2. write aour answers in the answer sheet prouided and shade th'e appropriate bubbles below your atuswers. 3. No steps are needed to justi'Jy Aour answers' I. Each question carries 1 mark. 5. No calculators are rl'llowed. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO 1

In this paper, fu.l denotes the greatest integer not exceeding r. For examples, L5l : 5; 12.81 :2' and l-2.31 -3. l. Find the sum 12 x 3f 22 x 4+32 x 5* 42 x 6+. . *202 x22. 2. In the following figure, ABC is a triangle and both ABC' and AB'C are equilateral triangles. Let O be the meeting point of lines CC' and B B' . Find IBOC in degrees. 3. Consider the function g(r) - A"2 + 8", where ,4 and B are constants. Assume that z, u are tn'o numbers such that g(z - 3) : g(u+3), and u - u f 6. Find the largest possibie value of A (g(u + u))'z + Bg(u + u). 5. Let a1,a2,. ' ' and b1,b2, ' ' be two arithmetic progressions such that or : 10 and bt :24. and that a1s6 * b16s - 2014 Find the sum of the first twenty terms of the sequence at * bt, az * bz, as 1- be, ' ' ' . The figure below (not drawn to scale) shows a triangle ABC wlth BC = 8 cm, BA: 4 cm and AC :2^/3I cm. The point M is the midpoint of AC. Find the iength of BM in centimetres.

6. Il r,y and z are real numbers satisfying the equ ation 12 +yz -l z2 - ry -gz- zt:27 ' find the r:raximum value of lY - zl' 7. The set ,4 is a nolr-empty subset of the set {I,2,3,4'5,6,7' 8' 9} with the prope-rtv - that wheneve r a e A, then 10 - o € '4 How many possible subsets A are there? 8. In the sequence . 2014 _, ihe .ly'ih term is ffi. Find ,N 9. A real-valuod function I satisfies the equal-ion f(t) I f (, ] ,) numbers r I 0,I. Find 24 x f(-3)' 123123 81 2 4'4' 4's's's' '9'16'16' Let rn be the smallest value of the function 3r Determine lrnl . 15 12 24 Ta'zs'%' '%' -- !6r+ h +T + x) r-7 for all real 10. Given that o : log2 3, b : - 1og+ 5 and c : - logz 3 * iog, 5, find the value of . tt -2 a'o'c 2" b" 2b2 I oc 2c2 i ob 12. 13. 11. 14. Let o; e {1, Find the number oftriples (a, b, c) such that a,6, c are numbers in the set {1' 2' 3' " '15} satisfuing the conditions a < b - I and b < c - 2' The first term of a.n arithmetic progression is an integer and the common difference is 2. If the sum of the first n tlrms (n' > 1) of the arithmetic progression is 2014' fin

1b. Let ABC be a triangle wiih a : BC,b: AC and c: ,48. Assume that 3o2+3b2 : 5c2' Find the value of cot A + col., B cot C 17. Find the number of integers A in the set ^g - {1, 2, 3, . . . ,1,20i.4} such that ft is of exactly one of the following forms 21 . n2,n3,n5, where n iu an integer. (Note: For example, 100, 1000 are such numbers but 64:82 - 43 is not.) Let a,b.,c be positive real numbers such that obc: 1. Find the least possible value of 201.4 2074 2014 a3(6 + c) ' 611o 1-.1 | cl, A A sequence ao,gt,az, a3, .. ., r.vith at:7, is defined such that for any positive integers rzr, and n, where m) n, the terms of the sequence satisfy the relation am+n I am,n + rn - n - irr^ + a2.) + J.. Find the uulu" of I o'otn l. L2074 ) Let ru be a positive integer such that, for each of the digits 0, 1,.. ., 9, there exists a factor of n ending in that digit. What is the smallest possible value of n? Let a1,a2,a3,... )a2o0t)... be an arithmetic progression such that al+ aloo, < t0. Find the largest possible value of the fotlowing expression atoot * (i,noz* rhss3 * . .. * azoor. It is given that w, a,b and c are positive integcrs that satisfy the equation l: a! +b! + c!. Find ihe largest possrble value of u I o+b 1.. In the triangle ABC, AB :63 cm,BC :56 cm.CA:49 cm, M is the midpoint of BC , and the extension of AM meets the circumcircle a_, of the triangle AEC at P. The circle through P and tangent to BC at M intersects o at e distinct from P. Find the length of MQ in centimetres. 18. 19. 20. 22.

24. Let M and C be two tiistinct points on the arc AB of a circle such that M is the rriap"i"i "f rft. *c AB. If D is the foot of the perpendicular hom M onto the chord AC such that 4p : 100 cm and DC: 36 cm, find the Iength of the chord BC in centimetres. Let AD bethe bisector of lA of the triangle ABC. Let M atd N be points on ,48 *ra iC,respectively such that IMDA: IABC and' INDA: IACB' Let P be tuu iot"rr."tion between AD and Ml{. suppose AD : 14 cm. Find the r'n}ue of AB x AC x AP in cm3.