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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) ?01? Senior Section (Rouna r) Tuesday, 30 May 2017 0930 - 1200 hrs Instructions to contestants 1. Ansuer ALL 35 wuti,ons. 2. Enter lJour ansuers on the dBurer sheet pnnid,ed.. 3. For the multiple chai.:e .tuesions, "nt", you, on".n o, ine ansuer sheet b! shadNng the babble containing the letter (A, B. a, D or Et .ateapan(lns to the c.frect ansuer. 1. Far the other short Westions, urite yaLr ansuer on the ansuer sheet and sha.le the opptopriat" bubbl" beto, lout o"suter. 5. Na steps are need.ed, to justify Aour ansuers. 6. Each question caffi$ 1 mark. 7. No calculatorc are alloued. PLEASE DO NOT TURN O\/ER UNTIL YOU ARE TOLD TO DO SO. Supported by l\,4 nistry oi Education Sponsored by l\,4icron Technology 6Qtcron

Multiple Choice Questions 1- Find all the \,alues oftr for which the exprcssion 12 +&(k r)+3h is positive for a[ x] (A) O

. Find the range of values of r ihat satisfu both of the followiag inequa.lities: (x + 3)2 x2. (A)0sr

19. r8. 21. What is the last digit .1201720171 22. 26 25 (,n 4cos (f) cos (+) cos (+),.* (T) - z-" (i) Tbere are 5 girls a.nd 5 boys in a junior class. and 4 sirts and 9 boys iII a senior class. ,{ conmittee of 7 m€mb€rs is to be formed by selecting studenrs from these two classes. Find the nunber of $'ays ihis can be done ifthe comniitee must har€ exacitv 4 s€niors and exactly 5 boys. Suppose r is a real number. Find ihe largest possible integer that can be attainei:i b). 777a t 101 l/ 5 r," ,i 1)' 23 In hov many ways can 4 int€gers, ar < .t2 1. Determine the ma-{imum vatue of az withh the sequence {a1. ar. a3,...} if bn+r - b, : 2foraUm >1,as:6tj and ato : 1045. There are 12 blue socks 14 re.l socks, 16 sreen socks, 18 yellow socks and 20 oranse socks in a drawer. Socks of the same colour lrxe indistinguishabte. A p€rson randomb, pi&s a certain nunber of socks from the drawer. Find the minimum number of socks that should be taken to ensu.e that he wiu hal,e at least two pairs of cotour J(. at leasl iwo pairs of colour y and ai least tllo pairs of colour Z, for some rltlee distinct cotouft x, Y, z. Determine the number of paths to morc from rhe toplefi celt ro the bottom-right celi in the 8 x 6 cell grid using a sequence of doi.nr,,ards moves ard right$',ards nov€s such that th€re nre an even number of direction changes.

Consider th€ following equation where m and n a.re positi!€ int€gers: 3-+3n-8m-4nl :680. Determine the sum of al1 possible values of rn. In the fi$rle below. ,4BCD is a parallelogram, where E ties on the srraight line Ce. The line tO intersects the [nes AC and Aj] at the point i. and P respectil€]y, r.here P is ihe midpoint of the line ,18. Let lAEl. IAD , IACI al.,d AF denote the lensth of the lire segment AE, AD. AC aud AF respectively. If lAt : 3 cm, 1,,{rl : 9 cm, fi",i 6 , lA'l \AF 29. Find the smalest positive integer n such that 5(8, + 2r)(34 + 24)(33 + 23)... (3r" + 22-) > 9156 30. The foltowing diagram shows paths (edges in the grid) connecting 5 x 4 lattice points. Each path is exactiy 1 meter long. Determine the shortest distance (in meters) a person needs to travel so ihai he will walk through each path at lea"st onc€ and rciurns to the sta-rting position. 27. 28. Let a. 6, :. and y be real va-riables such that ax+ba:4A, a; +bu2 =710, axi +by3:3t0, &t)1 +b!4:8gA DF,, rmio" bF v" u. a'oi - b'r'. 31.

Deiermine the largest prime number thai carnot be ereressed as the sum of three composite odd intesels. Two lines PQ and BS conneci the \,efiices of a regular decasor (lGsided polygon) ard intersect at the point ?. Suppose ZPIS = 2". Find r. In the figxre below, the lines,4B, AC and, PQ are tangent to the circle (with centrc O) at the points B, C and R respecii\,€ly. If the circle has.radiru 9 cm and the length of the line segment OA is 15 cm, find the perimeter of A,4PO in c r- In ihe {igure below, the circle centrcd at ,4 has radius 20 cn and the circle centred at -D has radius 5 cm. The lire segment QR is tangent to the smaler circle at the point P. Let 8Bl a,lld lBnl denote the lensth of ih€ line sesment 8B and BR respectivel,-. Find the ialue of IQB x lr,Bl. 34. 35.