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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2012 Junior Section (Round 1) Ttresday, 30 May 2017 Instructions to contestants PLEASE DO NOT TURN OVER 'irLteser l$s than or equal ta x. For 0930-120o hrs TOLD T'O DO SO Sponsored by l\4icron Technology L Answer ALL 35 Westians. 2. Enter your ansuers on the anslter sheet prol)id,ed. 3. For the multble chairc questxons, enter yoLr ansuer an the ansuer sheet bU shading bLbbLe containins the letter (A, B, C, D or E) cal-respan(tiw to the caryect ansuer. .4. Far the other shad questions, ur-ite your aneuer un the i".r,"" ,t"rt and shad.e the propl-iate bLbble behw your ansuer. 5. No steps are need.etl. to justutr gour ansuers. 6. Each question carr.ies 1 mark. 7. No ca.lculators are alloue.l. 6. ThmushoLt this paper, Iet lrl d.,nate the grEatest exampLe, l2.Il: 2, 3.9 = 3. UNTIL YOU ARE Supported by l\4inis1ry of Educqtion

Multiple Choice Questions 1. -{mong ihe flve umbe$ 25 26 46' +7 25 46 .2324(A) ,14 (Il ) 45 23 24 tl' E' (c) ulla fi, *ncr, o,,e t * the smallesi vatue? @) 2; (o) asl 2. Let o and b be real numbers satisfying 1 1 | a 6. .c(A) la b (C) a+b &3 (E) a2 > b2 3- How many ways can th€ letters of the word "IGt OO" be arradged? (A) 4 (B) 5 (c) 30 (D) 60 (E) 120 4. Jenny and Mary received identica.l fruit baskets, ea-ch containing 3 apphs, 4 oranges and 2 bananas. Assuming that both Jenny and Mary randomly picked a liuit fiom their own basket, wha.t is the probability that they both picked a,r apple? (A) ; (B) ; A cylinder has base radius r and height ?r. If a sphere has the same surface arca as the cylinder, find the ratio of the volume of the tylinder to ihe votume of the sphere. 1dt fA, " rB, ' rc, a ,o, "t 4J2r32 Let ABC D be a rcctangular sheet of paper with ,48 = 6 and BC : 8. We can fold the paper aloag the crease line t-P so that point C coincides with point ,4. Find the lengih of the resulting line segment ,4I - (c) ; 1l) ; (E) None or the above AED 5. (A) -27l( ) t (E) None of the abowe 25 ,*, ]! 42 (D)'7 7l Given tlree consecuti\€ positi\€ iateg;rs, whlch of the follorring is a pbssible ralue for the ditr€rence of iLe squares oI Lihe larycsl :r,nd the smallesi of ihese three iriegers? lA) e1 (B) g2 (c)' e3 (D) e4 (E) e5

9. Let a arld 6 be positive integels. If the highest co]I1mon facror of a and 6is 6 and the low€st common multiple of a and b is 233455, how many possible values a.re there for a? 8. You have 30 rods of length 5, 30 mds of length 17 ard 30 .ods of tength 19. Usiag each .od at most once, how ma.ny non-congruent tria.ngles can you form? (A) 6 (B) 7 (c) 8 iD) e (E) 10 (A) 2 (B) 4 (c) 8 (D) 14 (E) 16 10. Tf u and g are non zerc rea.l numberc saiistrillg x + g : 2 arrd find the value of rg. (A) i (B) -1 (c) Short Questions (D) t @) vA 2017r + 20772 +... + 2or12or7 1 t 1l_ An n sided polygon has two interior a.ngles of siz€s 94" and b1". The remainins interio, angles are all cqudl ixtu". ll 4. a _20 daFrminF r.F lallF o. n. Find the mrmber of multiptes of 7 ir the sequence 80,81,82,...,2016,2017. A list of six positi.!.e intege$ has a unique mode of 4, median of 6 and mea.n of 8. Find the lalgest possible inteser in the list. In the diagram, ,4F is a dianeter of the ctucle aJld ,4BCD is a square with points B and C on -4F and poinis A and D on the circle. If AB = 17.y/5 find the lensth of rF. 12. 13. 14. 15. Find tbe remainder when is diwided by 9.

16. Assume that lz)'!ai 1r,211016 lr i2,2ooo o,n1t2o- t o.0ro."2016 - ax+aa. F: d rhe valuF ol thF following Fxprpssion: (ao - or) + (az a3) + (aa a5) +. + (ozon ozon). 17. l' .r - ,"/2D17 l, frnd In" vdluc of x3 Q+ \O,OlTx2 + (1+ 2\4]017)r - 2U7. Let ABC be a t a.ngle, D be a point on Ad such that .4D = DC and E be a point on BC such that B-U : 2rd. Let I. be the intersection of BD and AE. If the area of tdangle ,4BC is 100, find the area of triafigle ADi'. Find the laxgest integer from 1 to 100 which has exactly 3 positive integer divisols. For example, the only positive divisom of 4 arc 1, 2 and 4. L€t d, b and c be positiw integers such that a2 + bc:257 arr,d ab +ln:101. Detemioe In a trapezium - BCD, AD is paralel to BC and poinis -A and -F arc the midpoints of 48 and DC respeo ir ely. Tl ArFaot AErD rttt Ar.a ot fB1-F 3 3' and the a.rca. of tdargle "48, is v/5, frnd the :rea of the irapezirm ,4BCt. Lg1 !al3:6a6,q,h6reaisapositiveintegerandbisatealnumbersatisfying0

I 28. 27. 29. Find the va.1ue of ar if . x5 : a,s(x - 1)5 + aa(r - 1)a+a3(e i)3+ar(r 1)'+o1(c 1)+a6. 31- Find ihe vAlrre of 32. Find the la.rgest possible value of ra such that the polynomial 12 1 (2n 1)r + (n 6) ha.s two rcal rcots ,1 and 12 satisfuing 11 ! -1 and rr ) 1: If one of the integers is rcmorred from the first N consecutir€ inteeers 1,2,3, .. . . N, the rFsu riDg d\eragF ot thp rpmanins ir rpse-s is ?. .'^O n. Let m be the mirimom value of the quadratic curve g : 72 4an + 5o2 3(1, where the \.alue m depends on !r. If 0 S a _< 6, find the maximrm possible .""!lue of m- Let a,b,c,d, anC, ebe fve consecutive positiF integ€E q here e is the largesh. Ifb+c+dis a pedect square and a + r + c + d + € is d perfect cubc, fi d tLc least possiblc \alue of e. 30. Let a and b be positive real numb€N satisfying a + 6 = 10. Find the largest possible .value of '/rr,a.+ts+'/tort+n. 34. Amongst the fractions 723 174 175' 1,75' !75" 175', there a.re some which can be rcduced to a fraction \vith a smaller denominator such as tfu : *1, and there are some that cannot be rcduced further like r75!. Find the sum of alt the ftactions vhich cannot be reduced further. 35- The number of seashells collected by 13 boys and n girls is n2 + 10n 18. If each child collects eiactly the same number of seashells, determine the !?lue of n. r 20 7V1+20r6Vr- 20rs,v/r 20rav4 -20R . 20 .