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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2012 (Open Section, Round One) Wednesday,3l May 2017 0930-1200 hrs Instructions to contestants 1. Ansuer ALL 25 questions. 2. Wite UoLr ansuers in the ansuer shpel praurded arul, shade the opp76\";o7" bubbles belou your ansuers. 3. No steps are needed, ta jlstif! lour ansuer' ,1. Each qu*tian cdrries 1 mark. 5. No ccltculators are allowed.. PLEASE DO NOT TURN OVER UNTIL YOU AR,E TOLD TO DO SO Sponsored by lvlicron Technoloqy Supported by Minislry ol Educaiion 1 4fric.ron"

In this paper, let R denote the set of all real numbers. Let Ljdl denoie the geatest integer less than or equal to c. For examples, 5 = 5, 2.81 = 2, a.trd ]. 2.3] : 3 t. Let r be a real numt". ".,"u *'ut s: .,/iF + f4 - (t +r)201? is an int.ger. Fintt the units disit of S. 2. Find the value of ----= - _----------: Vr7 '/288 - /7 y'48 V'7 /288 - V7 r'48. 3. Ifz:+*tf";f;* 2Ol7 x T. ++ 1+2+3+,1 1+2+3+4+.+2016' 4. Find the !a1ue of d srrh ihat / lrld' : 55. m r:4i+8j 3k,+ /tr(2i j+2k), r€lR, respectiveiy. The line with equation 5. In ihe three dimensional space iith odgir O, let i, j and k be thrce mutualLy perpendicular unjt lectols defined in the us al aniiclocklise s€nse. Ts'o intersectinq lines I and rn have t: r=i+2j+3k+.1(i+2j 2k), ,\€R. and 6nd r = i+dj +6k+u(ci+j), ur € R and m and bisects the angle subtended by I ald m. Find the l€.lue Find the lrrlue of o'? + 2y'7 + 2a. intersects Ure lines I of a+6+c. Leta:vr Lv/1. 6. 7. In how many ways can 2017 be e4ressed as the sum of one or more positile idtegers in non-decreasing order. such that ihe difer€nce betscen the last iernl and the firsi telm iq at most 1? In the given quaddlateral ABCD, BC = CD: BD. AB:6,AD: a and IBAD:34' Find.4C.

9. Gi!'e{ that in triangle ,4BC jn which ,18: AC and lB,4C = 120", , is the point on BC such that DD : l0 and DC : 20. Find .,1D. 11. to_ In the isosceles tdangle ,48C, AB : ,1C : 40, BC : 20Jj, and p is a point or the segment ,4-B. The circumcicle of ihe tdargle pBC intelsects the segment /d at q. Suppose the idangles,4PQ ad PBC have the same circunnadius. Find the lengrh of AQ. Let a. 6, c and d be posiiil€ numben such rhat a + b + c + d : 8j ard thai t- a J- o-" t, -a "tt,." i t'r6 1," ro,," n" , ou , 80 - qo -ao a). t oL t.J o b d o.b_c / z ,a\" 12 ret.n: | ; - , i l . where ;: + 1 : 0. Find the least positive inteser n such that \- '/ 1."-1 :"2>zooo. 13. L€t A,4BC be a triansle $ith a : BC, , = AC an

L 19. For ary positiv€ integer n, let an be the units digit ofthe sum 1+2+3+ +n Thus at-1)a2=3,a3:6 aa=0 Determine o' for r = 20172017' 20. Let a0:5 and a'+ra" = ol + l for all n 2 0 Determine la1666l' 21. Ho -many 7-digit numbers formed by using onlv the digits 3 and 7 and diwisible by 21 are there? 22. Each cell of a 20x20 table is painted with a colour so that the cells in each row and each collllnn a.re coloured by at most 7 colouls. Determine the ma;dmurn number of coloum that can be used. 23. Let ct1,a2,aB,tLa, -be the sequence formed bv adding the nurnbers jn all the possible no.reE o,y Coir- "rb"'." or lJo 3r,3/ 3J - and arF "rrdn8ed a d r,lonororF in'reasug "0".*". rn*.o'= , io u, -3'- Jria" 4l--30 3'u' -9-3') o'-t - 30 + 32),. and so on. Determin€ the ralue of 421?. 24. Let (a^) be a sequence $'ith a16 :2',4 and - : '--* for n z 2 Find the lalue of 25. In a triansle ABC, AB: AC, BC = 22'"6. cosA = +, Dk a point onlC such thai ,4, < ,C and P is the point on the segment BD such that lAPd = 90"- Given .t ABD: lBCP,1td BD.