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

Singapore Mathematical Society Wednesday, 1 June 2016 Instructions to contestants 1. Answer ALL 25 questi,ons. 2. Wr.ite yaur dnliuers in the anslner sheet pratid.etl. and sha.d,e gollr ansuers. 3. No ste+s are need.ed to jus rtJ Uour.tnsuers. 1. Ea.ch questian cdrr.ies 1 rnarh. 5 \a rc|",tot^r' orP o!lauted. Singapore Mathematical Olympiad (SMO) 2016 Open Section (Round 1) the appropridte bubbles belou 0930-1200 hrs Sponsored by Micron Technology PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO Suppqted by Minislry of Education 4fric.ron'

In this paper, let l$l denote th€ greatest intcger not l1\ceediru a. For c,xa,.npt€s, l5l : i, i2.81 : 2, and | 2.3J : 3. 1. The perimeter of a triangle ABC is 48. The point D is rhe ntidpoint of r4B suc.h that DC : DA: fi. Fi.rd thc arca of trjansle CBr. 2. In an infiniie geometric prosressiox with a nonzero tirst tcrm al]d a common ratio :, . the sum of ihe first n terms equals the sum of all the remaining terms. Fincl n. 2 3. Find l,he dilTerence betw€en the largest and smallest aahe of r rvhich sarisfies ihe equation l,-2 r 20161 :2lr- 1009. 4. The ligure belov shows a 10 x I rectansular board. Atl the snal squares sho{.n h ihe figure ar..e squa.es of ihe same size. Find the total rmmber of rcctangles in ihe above figure vhich arc not squares. 5. Find the miDinlum \,?lue of the function f siv€n by t..,1- rG 2b -,f,Bo , rG 6. Lct m be ihe number of those triangles shose iongest side is 2016 and the other two sides ar. aLo oIinLefla lensrt D.rermine I n t- - 100(,.1 (Note: Two congmeni triangles are considered to be the sane triangte.) ls 7. rind l2 Fh-- -71 1 - ,1 ,2 "'l oo

a (Hint: Nole that uE + JFi t/i + JTi' 8. Fnld the number of lnteger solutiotrs to the equation ./1 ,,G vOOlE c + .,6 1,5 JmE r e. Let sa = i. *. n"*rL * value of 2017 x S:mre . ++ 1+2+3+4 1+2+3+ +N 10. 11. 12. 13. 1 1009 r - 1009 Find the Let i,j and k be thr€e unit v€ctors along three mutually perpendicular a-aes. nalnely, the z, g :nd z axes rcspectively, and the orisin O is ihe inte$ection of the three a-{es. At any time I > 0 after the start of an er,?edment, the position of a ioy plane A is locaied along ihepathr=i+2j+3k+t(Si+5j+2k) ard thc position of another plane.B is located alongth€pathr:1i+7j+2k+tl2i+3j+ak). If d is the shortest possible distance betrl'een the trv'o toy planes, find d2. Let i,j and k be thrce unit vectors along three mutually pependicular s-xes, namely, the ,, g/ and z ax€s rcspectivel)'. ard ihe otigin O is tlie intersection of the ihree a-{€s. A plane milror in the space has equation r'(i+2j + 3k) = 5. A ray of licht is shone alons the path with equaiior r: 5i+2j +k+l(5i+j), where ) is real. The ray of light hits the plane and is reflected along ihe paih r : j + k + p(4i + bj i &), where p is real. Assumins that the incideni ra], reflected ray and the normal at the point of i cidence lie on the samc plane, and that the incident ray and the rcflected ray make the same angle with the nirror, frnd ihe \.-alu€ of Dl + lc . Cjven ihai 2:1;,,,:l1a 6,.a l ::Y1l a-o a Lrl Findthenumberofwaystoselectfourdistinctintegersa,i,,canddliorl{1,2,3,4,..,24} such that a >, > d > c and that o+ c : b+ d. 14. Assume that I :',/tar+ 1+ \/tar+1+ v4o: + t. . + v€cro.- + t \i'here al,o2, , d20r6 are real numbers such that al + d2 + . . + a2ar6 = l Find the m:ximum value of l/1. ,'1d Let p and q be intesers sucl! that tt{Jqgls of the polinomiat l(r ) = i + px2 + qt 343 ' axe real. Determine ihe minimujl.I possible value of ll - zg . 16. Iet /(r) :,2016 + ozots"2ar1 + a2614x2414 + + alrr +a0 be a polj.nomial such that f (i,):2i: lrotaIIi= 1,2, ,2015. rtud ihe value of /(0) + l(2016) 20161.

T' 17. Find the laxsest prine number ! such that p : a' + 7ba holds for prime numbers a ar.] 6. 18. Fjnd the remain.ler lhen 1 x 3 x 5 x . x 2017 is divided by 1000. 19. In ho{.nany ways car the rumber ;3H be writien as the p.oduct of two fractions of the form *, wherc d is a positive inteser? In*" i,p!# ad ++ are consldered as the same product.l 20 In the idargle ,48C, lB :9Ao nnd lC : 60". Points , and t are outsid," the triangle ABC such that 34, and ACt are equilaterai trisngles. The segmeni tt intersecls the sesment ,4C at ]r. Suppose ,C : 10. Fnld the lensth of ,4F. A circle r",r of radius 8 is intertrally targent to a circle !,r2 of radius 25 at a point T. A hre through the centre O of u2 is tangent to {r1 at S. A chord AB of u2 perpendicular to OS is tansent to {r1 at Q. Find the lensth of ,.1-B. In a rectansle ABCD, E and -E are the miclpoints of BC and CD respectiveLr-, tE intersects,4-F at P, ,E iniersects Bl'at 8, and.4.E inteNects B-P at B. Giren thc area of the tdangle PQR equals 100, find thc a.ea of thc triansle ,88. Let ABC be an acute angled tdangle q'ith cfcumcenter O. Suppose AC : 92, tar lO'4, : + a d ICAO = 3IOAB. l'ind the length of ,48. In the triargle ,48C, lC = 90'.2AC > AB, points -E and ,F' on AC and ,4, respecti\€ly axe such ihai CE : -E.F.. Suppose EF is the segment of minimal length that di\,ides the area of triadgle ,4EC into two equal halves. Given /C : 6 + 3./6, find the iength of ,,{B. Determine the greatest positive constant a su.h that lf +s'-2>a'\!+! r 2) lor alls>0. 22 23. 21. 25. 24.