Download [PDF] 2014 SMO Junior Section Round 1 Questions Singapore Mathematical Olympiad

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Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2014 Junior Section (First Round) I\-resday, 3 June 2Oi4 fnstructions to contestants 1. Ansiter ALL 35 questions. 2. Enter Uour dnsuers on the answer sheet prouided. 3. For t'he multiple choice questi,ons, enter your ansaer on the ensuer sheet by shading the bubble containing the letter (A, B, C, D or E) cotresponding to the correct ansuer. ,1. For the other shoft questions, wrile your anstrler on lhe answer sheet and. shad.e lhe ap_ propriate bubble below your qnswer. 5. No sleps are needcd to juslify your ansuers. 6. Each question canies 1 marh. 7. No colculalors are ollow.d. 8. Throughout this paper, let lr) denote the grcatest integer ress than or equar to x. For exampte, L2.11 :2, l.3.9J : 3. PLEASE,DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SO 0930-120O hrs

Multiple Choice Questions l.Letn,yandzberealnumberssatisfyingr>y>0andzl0.Whichoftheinequalities below is nol, always true? (A) n+z>y+z (B) c-z>a-z (Q) rz>vz (D) !+z>f,+z (E) rz2 > gz2 2. If the radius of a circle is increased by 100%, the a.rea is correspondingly increased by how many percent? (A) 50% (B) 100% (c) 200% (D) 300% (E) 400% 3. If a : rt, b: '/do, find the value of ./63. (^ J:: B b-:" (c) .$ (D) * (E) Noneof rheabove vrl "r/tg \-t l0 b . , t00 4. Find the value of --L + l--: + -] -. '*'*" "' I - ltB' t1Y5 ' t+*6' (A) -1 (B) 1 (C) -/5 (D) r'5 (E) None of the above 5. Andrew, Catherine, Michael, Nick and Sally ordered difierent items for lunch. These are (in no pa.rticula.r order): cheese sandwich, chicken rice' duck rice, noodles and steak. Find out what Catherine had for lunch if we are given the following information: 1. Nick sat between liis friend Sa.l1y ard the person who ordered steak. 2. Michael does nol likc noodles. 3. The person who a,te noodles is Sally's cousin. 4. Ncither Catherine, N{ichael nor Nick likes rice. 5. Andlew had duck rice. (A) Cheese sandrvich (B) Chicken rice (C) Duck rice (D) Noodles (E) Steak 6. At 2:40 pm, the angle formed by the hour and minute ha,nds of a clock is oo, where 0 < c < 180. What is the value of c? (A) 60' (B) 80' (c) 100" (D) 120' (E) 160" 7. In',the fig1re below, each distinct letter represents a unique digit such that the arithmetic sum hoLds. If the lettel L represents 9, what is the digit represented by the letter T? TERRIBLE +NUMBER THIRTEEN. (A) 4 (B) 5 (c) 6 (D) 7 (E) 8

8. A regular cube is to have 2 faces coloured red, 2 faces coloured blue and 2 faces coloured orange. We consider two colourings to be the same if one can be obtained by a rotation of the cube from another. How many different colourings are there? (A) 4 (B) 5 (c) 6 (D) 8 (E) e 9. It AABC, AB: AC, IBAC :120', D is the midpoint of BC, arld. E is a point on -48 such that DE is perpendiculax to.AB. Find the ratio AE: BD. (A) 1:2 (B) 2:3 (c) t:y'5 (D) t:2t/5 (E) 2:3J-J 10. How many Eays are there to add four positive odd numbers to get a sum of 22? (A) 14 (B) i5 (c) 16 Short Questions (D) 17 (E) 18 11. Succcssive discounts of 10% and 20% are equivalent to a single discount of r%. What is thc value of r? 12. The diagram below shorvs the front view of a container, with a rectangular base. The container is filled with q,-ater up to a height of 6 cm. If the container is turned upside dorvn, tbe height of the empty space is 2 cm- Given that the total volume of the container is 28 cm3, find the volume of the rvater in cm3. l 2 cnl, i ,"^ 73. Le't, A be the solution of Find the value of 61. the equation r-7 z-8 r-lO :r-8-n-9:"-n r-12' 14. The sum of the two smallest positive divisors of an integer N is 6, while the sum of the two largest positive divisors of N is 1122. Find N. 15. Let D be the absolute value of the difference of the two roots of the equation 3r2 - 70x - zOr :0. Find [r]. n-77

16. If rn and n are positive real numbers satisfying the equation m + 4\/tnn - 2\/tn - 41/A + 4n:3, find the value o, t/m + zt/i + zo]4' '' 4 '/*-2'/i 17. In the diagram below, ABCD'tE a trapezium with.4B ll DC afi, IABC = 90o. Points E and F lie on .4.B an

22. Let ABCD be a square of sides g cm. If E and F are la.riabre points on BC and cD respectively such that BE: cF, find the smallest possible a.rea of the triangle x^q,pp io cm2. If o,6 and c a.re n "i;Trfrrl**bers satisfying a* 2b-flc:20t4 and2a+Jb+2c:2074, find the valrre ac*bc-ab In the diagram below, AABC ":d lqDp are two right_angled triangles with AC :24, CE:7 and LACB : 4CED. Find rhe length of thJline sfiment.,{.8. The hypotenuse of a right-angled triangle is 10 and thb ra.dius of the inscribed circle is Find the perimeter of the triangle. Let r bea real number satisfying (o - i)' : 3. Eva.luate ", * *. For 2 S c < 8, .*" d"lig l.@) : b - 2l + lr - al - pr - 61. Find the sum of the ta.rgesr and sma,llest vatues of f (c). If both n and t/&T 2dE are positive integers, find the maximum value of n. Let N :dd,be a digit perfect square that satisfies 6: s.ca+ t. Find the sum of all possible l'a,lues of N. (The notation n : dD means that n is a 2-digit number and its value is given by n : 10o*6.) zJ. 26. 27. 28. to

30. Find f[g fJ'llowing sum: (;.i.i.i* .*). (?.i.;. .*) . (:.;++ +*)- * (,,2*';) -'j. 37. lf aa I fu : 7, ax2 + W2 : 49, ars + W3: 133, and oro + bgo= 406, find the value of 2oI4(x + a - ca) - 100(o + b). 32. Fot a > $, we define a+l 3 Find ihe maximum ralue of 9(a). 33. In the diagram below, ,4D is perpendicular to ,4C and, IBAD : IDAE: 12". If AB + AE : BC, frid, IABC. 34. Define ,9 to be the set consisting of positive integers n, such that the inequalities 17< "+k< 15' hold for eractly one positive integer k. Find the largest element of ,9. 35. The number 22e has exactly 9 distinct digits. Which digit is missing? 18"-l Vr +