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*< Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2015 (Open Section, Round 1) Thursday, 4 June 2OL5 0930-1200 hrs Instructions to contestants 1. Answer ALL 25 questions. 2. Wri,te your,answers i.n the answer sheet proui,ded a,nd shade the appropriate bubbles below youT answeTs. 3. No steps are needed to justi'fy your answers. /1. Each quest'i,on carri,es.l mark. 5. No caLcuLators are allowed,. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO

In this paper: let lr.l denote the greatest integer not exceeding r' For examples, [5-J : 5, L2 8J : 2, and L-2.31 - -3. -'"1t'pirra the largest positive integer .l./ for which ,5 -5r3 *4n rs divisible by N for ail positive integers n. 2. Consider all sequences of numbers with distinct terms which follow a geometric progression such that the fir-st, second and fourth terms of the sequence are three consecutive terms of an arithmetic progression. Find the sum of squares of all the possible common ratios of these sequences. 3. Srrppo." that a given sequence {r,"} satisfies the conditions that rt -- and, for n ) t, Ln*r:ltt* 4rn* r+t6;) lb' Determine ,l$ 3r',. $,.In tbe figure beiow, E is a point inside the paralielogram ABCD such that IDAE : -/ / DCE: 50o and IABE: 30o. Ftnd IADE in degrees' -'C' 111 1 rt is given that ,9 -- #+ * + t; +. . .+ ^G:,C. Find the value of [4030 x S.J. Let ABC D be a trapezium with AB parallel to DC, and that AB : 20 cm, c D : 30 cm, BD:40 cm and. AC:30 cm. Let E be the intersection of AC and BD. Find the area of triangle DBC in cm2. 6. : \ i 7. How many distinct integers are there in the following sequence: I t2 I I 22 I I J2 I l2oLs2l, Lr*l 'L**l ' Lrttl '"'' I zorr l' .g:'Let ^9 be the sum of ali the positive solutions of the equation Find [S]. 4-12 -a ,J 12 -L J -1.

fi + 10)(1 + 102)(1 + 104) ... (1 + toz-; r-9. Given thut - 1, find the va,lue of rn. 1 + 10 + 102 + 103 + 104 +... +I0t2z J0. Let a,-be the nth ierm of a geomeiric progression, where at:I and a3:3. Find (X fll''*') (,4'-"'(T)',.') 14. 15. 11. Let g : J3Br - I52 + t4du -6Ii be a real function. Find the largest possible value of 12. Find the coefiflcient of r50 in the expansion ot (r+7)(r+2)(r+3) ...(z+50)(r+SI)(r+52). k, |, i'{ ,!& Fifty numbers from the set {1, 2,. . . ,100} are chosen and another fifty numbers frorn the set {101, I02, . .. ,200} are chosen. It is known that no two chosen numbers differ by 0 or 100. Determine the sum of all the 100 chosen numbers. Let H: (rt -*2 +r)e, *here ,: -]-. Determine LIIJ. {5-l L Assume that r2y 2 "2 # and r -la*z: t.Let M arrd mbe the largest possible value and the smaliest possible value of cos r sin gr cos z respectiveiy. Determine the value ^tlM I - ()t I - t- tm. l Let f : IR. --+ R. be a function such that, for any r,.r7 € )R., (r - y)t(r + a) - @ + y)f(r - a) : (6r'a + 2y\@2 - y2). Suppose f(1) : -999. Determine the value of f(10). 17. Let b,c,d and e be real numbers such that the following equation 15 - 2ora + br3 + "r, + d,r I e : o has real roots only. Find the largest possible value of b. 18. Let N denote the set of all positive integers. Suppose that f: N -+ N satisfies (u) f(1):1, (b) 3t(n)f(2n + 1) : t(2n)(I + 3f(n)) for ail n € N, (.) f(2n) < 6f(n) for all n e N. Defermine f(2015). 16.

19. Let T1,T2,fr|,fi4 and 15 be positive inlegers such that 11 * 12* rZl rSt rS: Il 12'u3'14':I)5. Find the largest possible value of 15. Ah Meng is going to pick up 2015 peanuts on the ground in several steps according to the following rules. In the first step, he picks up 1 peanut. For each next step, he picks up either the same number of peanuts or twice the number of peanuts of the previous step. What is the minimum number of steps that he can complete the task? Determine the number of integers in the set .9 : {I,2,3,.- ' , 10000} which are divisibie by exactly one of integers in {2,3,5,7}. 22. Determine the largest lnteger n such that n n-7 \--? -t - \--. ,?*' for all real numbers ,t1) fr2,. . .,rn. A circle cr1 centred at 01 intersects another circle cd2 centred at 02 at two distinct points P and Q. Points A and B are on 0J1 and w2 respectively such that AB is an external common tangent to a1 and w2. The line through PQ intersects the segments AB and OtOz at M and ly' respectively. Suppose the radius of c..r1 is 143 cm, the radius of. u2 is 78 cm and O1O2:169 cm. Determine the length of MN in centimetres. Let XY be a diameter of a circle ar of radius 10 cm centered at O. Let A and B be the points on XY such that X,A,O,B,Y are in this order and AO: OB:4 cm. Suppose that P is a point on cu such that the lines P.4 ar,d PB intersect w at C and D respectively with C and. D distinct from P. Given #: f , O"*"r-ine the ,atio Pfi. In a triangle ABC, the incircle c,., centred at l touches the sides BC, CA and AB al D, E and F respectively, Q is the point on cu diametrically opposite to D, and P is the intersection of the lines FQ and DE. Suppose that BC : 50 cm, CA:49 cm and PQ : Q.F. Determine the length of. AB in centimetres. 20. 27. 23. 24. 25.