[PDF] Senior Section - First Round - SMO Singapore Mathematical Olympiad 2016.pdf | Plain Text

Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2016 Senior Section (Round 1) Tuesday, 31 Mav 2016 0930 1200 hrs Instructions to contestants 1. Ansuer ALL 35 questions. 2. Enter Aour ansuers an the ansuer sheet prol)id,ed.. 3. For the multipLe choice Elesti,ans, entergaw ansuer on the answer sheet by shading the bubble containing the letter (A, B, C, D or E) com.spandtns ta the coffect .1. For the other short questions, u.ite go',r,r.tnsuer on the ensuer sheet and, shatl,e the awropriate bubble belou llouT ansuer. 5. No steps are needed ta justifu yaLr ansluers. 6. Each question carries I mark. 7. Na calatlala|s aIE alLoued. 8. Thraughout thLs paper, the constant e is the base oJ the natum,t Logartthm In. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO. Supported by lVinislry of Education Sponsored by Micron Technology 5fli

Multiple Choice Questions 1. If a, B are two distinct roots of the equation 2jc2 + 4, 45 : 0, 6nd the va.tue of a2 13 + dp2 . (A) i5 (B) 15 (c) 45 (D) 45 (E) eo z. Simptrfy 9; 9l E4 "1. tA, 3:l rBr 3{2i ,cr 3j2-i ,n, 3j2i ,., 3:2i 25 5 " b ''' s 3. Given that e2" e! : e2 and tlr(z + 29) = ltr 5 + ln 2, find the mtue of z + g/. {Ai2 (B) l {CJ 8 rD, t0 ,Et t2 4. colve lor r in tbF lo o$,ing "qua,tion loge. , loSr{27r, J0. (A) 3'6 (B) J'" (c) 3'" (D) 3'n (E) 3ro t 1,]^*]n"! "o.a:..12, where 0. ! a ! 90., and that ranB: !, where 180. < p < 270.. Find sin(B - a). ,A, :: (Br -!: rcr l! rn, 59 ,r. rJ ob 65 65 65 ,", 65 6. Which of the following is the greatest? tt / t \l/6 /A,/* ', (,is) '" ,",(,!) ,",(j)'"(1,) " ,E, (l),./,) . ,8 / \3/ 7. Find al uhe posir,re va rres ol r lor whrch 2lr2 Jxl

E (A) r:55', 135" (B) ,:75", 125' (C) z = 105', 165' (D) ': e5', 145' (E) r: 115', i75' L Which of the folloving is equal to .,,-2000 .2004 ! 2004 v4008 r/=zooa r zotz .7, "6n+"6G ( s\41 E.tE (B) 4./i - b'E (q 3,/14 - B'5 e) 3fi4 B,rE @) 3rt ./E 10. Let a : cos 282", 6 : cos 349', c : sin 102" arld d : sin169'. Which of the foltowrng is true? (A) b> a> c> d. (B) 6> c > a > d (C) c> a > d > b (D) a > tl > 6 > c (E)a>c>r1>b Short Questions 11. The expression 323 +Ax2+Br 10, where ,4 and B are integers, is divisible by 3r - 1 but leaves a rcmaindd of 14 when divided by jc+2- Find the value of ,4+8. 12. Find the laxgest positive integer p such that 12 + 2(1 +2p)x 2p 31 is a.lwa),s negative. 13. Find the ]argest integer smaller thar (2 +.,,/2)4. 14. Civen rl'a, r t 0. a.rd 5' I I !. rr"a ,hp value ot2j 5'2 i5. Find the largest value of z satisfynrg the following equation: .,/2, s7 +.,/" r=a. lb. Find rh- nunher ofsou: onc tor,hF follow ng "quarion: 2sinr + 1 :2lcosrl, where0"!a1360". 17. Find the ma.ximum value of 6cos2o 24sinlcos.r 4sin2r.where0"

19. 18. Given that tand: { and tanB : 3 Find the r",lue of 3sirl(d + P) 6sin a cos P {,,- 1lr, n t-)'o /'./ r/ 21. Find the smallest int€ser * > 23 such that is a positive i4t6ger. 22. Suppose r and 3/ axe two rea.l numbers such that 23. 2 sin a sinB + cos(@ + B) Consider the function f('):3"+21+31'l -lr 1 sla-:1, where r is any real number. Suppose A a"rd B a.re the maximum and minimum 16.lues of l(r) respectively. Find the vatue of A B. Find the coeffcient of ra in the expansion of I ., 20. 24. r+g =6 ard 2i +3Ea +a2 :12. Find the value of 12 + g2. Suppose a ciicle C is centred at the point (3, 1) on the rA plane, alld the line 43/+3, : 63 is tangent to the circle C. Find the radiDs of C. In the tria.ngle ,4BC below, IABC :2IACB, and ,4D is perpendicula.r to BC. Sup pose E is the midpoint ol BC, and DE :3 neters. Find the length of AB in meter. \ k2 - 23k

25. Suppose r and 9 are positive real numbers such that i < z + g < 9 anct u < 29 < 3r. Find the la.rcest value ot 9p I r' 26. The figure below shows a quadrilateral ,4BCD oD the ag-plane, where A(a,a I0), B(10,0) and d(0,5). The line .4,B is paxatlet to the tine Cr, and rhe tine ,4, is per pendicula.r to Cr. civen that a is a positive integer greater tha.n 10, find the smallest value of a such that the area of the quadrilaterat ,48C, is geaber than 200_ A(a a \O) c(0, 27. In the 6gure below, ,4-B is parallel to C, , AC = AB + C D and, -E is the midpoinr of BD. SLtppose IACD = 68'. find the angl€ ICAE in degree. 2E. Suppose ancalc is the smallest 6-digii riumber which is divisibte by 2016. d:8:.. a. b and I ecl , o bc dt n' L Find I F:l Ligj. umbF- 06.. Find the largest posibive integer n such that 19" divides 20161_ 29.

30. A sequence a1, 02, a3, ,. . satislies an]3 = 20,ny2 2a,,.4 I an for ?r : 1, 2. 3, . . .. If ar : 11, a:t :222 and q = 7777, lllrd a2n|l. How many ways are tlere to a-rrange the tetbers of ihe word ,RECURRENCE'in a row so thai no two R's a.re adjacent? The roads of a town form a norlh-by-east grid as sho$,n in the figrue below. Find the nurnber of ways a vehicle carl go from point O to point .P wherc onlv easteriv and northerly direciions are allor€d and no backtrackings are allowed. 31. 32. 1 o 33. In the following diagram, ABC, is a squaxe $.ith ,48 : 10 cm. Let Ar,Bl.Cl,Dr be points on the sid€s of ,48C, such that ,4,4r = BBt : CCt : OO, : j,lB. Similaxly, let A2,'2,C2,D.2 be points on the sides of ,41Brcrt1 such thar ,41;{2 : l BrB2: CrC2 - DrDz : ;ArBr. Bcpeat this proce.lure to construct infinitetv lrary b ABCD, A1BrCIDr, A2LJ2C2D2, .... Find the sum of their areas in cm2.

34. In the figure below, the point O is the centre of the circte, the tine BC is perpendicular to the line ,4E, a]rd the lin€ dD is perpendiculsr to the line ,4-B. If the radius of the cjrcle is 10 cm, find the vatue of OD2 + CD2 in cm2 35. Let o.r, a2,. . ., o16 be a.n arrangement of the numbe$ I,2,3,4, 5,6, Z, 8, 9, 10 such Lhs I (i) a1+ ct2 + o4: aa a 45 + a6 : 117 + as + ae,.and (ii) the number a16 is even and not equa.l to 10. How ma.ny such a,rrangemeDts are therc?