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Singapore Mathematical SocietySingapore Mathematical Olympiad (SMO) 2006
(Senior Section)Tuesday, 30 May 2006 0930-1200
Important:Answer ALL 35 questions.Enter your answers on the answer sheet provided.For the multiple choice questions, enter your answers in the answer sheet by shading thebubbles containing the letters (A, B, C, D or E) corresponding to the correct answers.For the other short questions, write your answers in answer sheet and shade the appro-priate bubbles below your answers.No steps are needed to justify your answers.Each question carries 1mark.No calculators are allowed.
1. Let p =23009, q =32006 and r =51003. Which of the following statements is true?(A ) p < q < r (B ) p < r < q (C) q < p < r (D ) r < p < q(E) q < r < p
2. Which of the following numbers is the largest?(A ) 3020 (B ) 1030 (C ) 3010 + 2020 (D ) (3 0 + 10)20 (E ) (3 0 x 20)10
3. What is the last digit of the number
(A ) 2 (B ) 4 (C ) 6 (D ) 8 (E ) 01 14. Let x be a number such that x + - = 4. Find the value of x3 + -3.X x'
(A) 48 (B) 50 (C ) 52 (D) 54 (E) None of the above16
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F
35. Consider the two curves y = 2::r3+ 6x + 1 and y = -2in the Cartesian plane.xFind the number of distinct points at which these two curves intersect.(A ) 1 (B ) 2 (C ) 3 (D ) 0 (E) 5
6. In the following figure, AB is the diameter of a circle with centre at O .. It isgiven that AB =4 cm, BC =3 cm, LABD =LDBE. Suppose the area of thequadrilateral ABC D is x cm' and the area of DDC E is y crrr'. Find the value of. xthe ratio -.y(A ) 7 (B ) 8 (C ) 4 (D ) 5 (E) 6
E
7. Five students A, B, C, D and E form a team to take part in a 5-leg relay competition.IfA cannot run the first leg and D cannot run the last leg, how many ways can wearrange them to run the relay?(A) 74 (B) 76 (C ) 78 (D) 80 (E) 82
8. There are n balls in a box, and the balls are numbered 1,2,3, ... .n respectively.One of the balls is removed from the box, and it turns out that the sum of thenumbers on the remaining balls in the box is 5048. If the number on the ballremoved from the box is m, find the value of m.(A ) 1 (B ) 2 (C ) 3 (D ) 4 (E ) 5
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9. Suppose a, b, c are real numbers such that a + b + c =0 and ab c =-100. Let1 1 1 .x = - + - + -. Which of the following statements IS true?ab c
(A ) x>O (B ) x=O (C) -l
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. . .
16. Evaluate
17. In the diagram below, ABCD is a square. The points A, Band C are collinear.The line segments AC and DC intersect at E, and the line segments DC and BCintersect at F. Suppose that DE=15 ern, EF=9 ern, and FC=x ern. Find thevalue of x.
EF
A~----------~------~GB
18. Find the sum of the coefficients of the polynomial
19. Different positive 3-digit integers are formed from the five digits 1, 2, 3, 5, 7, andrepetitions of the digits are allowed. As an example, such positive 3-digit integersinclude 352, 577, 111, etc. Find the sum of all the distinct positive 3-digit integersformed in this way.
120. Find the value of - 4 sin 70.sin 10121. Let w=1+ ~ + W + W + {ff6. Find the value of (1 + _ )30.W
22. Suppose A and B are two angles such thatsin A + sin B = 1 and cosA + cosB = o .
Find the value of 12cos 2A + 4 cos 2B.19
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23. Consider the SOO-digit integer2345234523452345.
The first m digits and the last n digits of the above integer are crossed out so thatthe sum of the remaining digits is 2345. Find the value of m+ n.
24. Let a and b be two integers. Suppose that V7 - 4J3 is a root of the equationx2 + ax + b = O. Find the value of b - a.
25. Suppose x and yare integers such that(x - 2004)(x - 2006) = 2Y.
Find the largest possible value of x + y.1
26. In the following diagram, LACB =90, DEl_BC, BE =AC, BD = 2 ern, andDE + BC = 1 cm. Suppose LABC = x", Find the value of x.A
D
B~------~~--~CE
27. If 1] (x) = ijx2 + 2x + 1 + ijx2 - 1 + ijx2 - 2x + 1
for all positive integers x, find the value of](1) + ](3) + ](5) + ... + ](997) + ](999).
2S. In the figure below, S is a point on QR and U is a point on P R. The line segmentsPS and QU intersect at the point T. It is given that PT =TS and QS =2RS. Ifthe area of D.PQ R is 150 em 2 and the area of D.P SU is x em 2. Find the value ofx.
p
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29. Let a and b be two integers. Suppose x2 - x-I is a factor of the polynomialax5 + bx" + 1. Find the value of a.
. J 6 - v 1 .30. If sm e - cos e = 2 ' find the value of 24(sm3 e - cos" e ) 2 .
31. How many ordered pairs of positive integers (x , y ) satisfy the equationxVfj + yVx + J2006xy - V2006x - J2006y - 2006 = O?
32. Find the remainder when the integer1x 3 x 5 x 7 x ... x 2003 x 2005
is divided by 1000.
33. Let 1 : N ----t Q be a function, where N denotes the set of natural numbers, and Qdenotes the set of rational numbers. Suppose that 1 (1 ) =~,and
for all natural numbers x, y. Find the value of 1(20).
34. Suppose xo, Xl, X 2, ... is a sequence of numbers such that Xo =1000, and
for all n 2: 1. Find the value of1 1 2 997 99822 X o + "2XI + X2 + 2X3 + 2 X4 + ... + 2 X999 + 2 XlOOO
35. Let p be an integer such that both roots of the equation5X2 - 5px + (66p - 1) = 0
are positive integers. Find the value of p.
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Saturday, 24 June 2006 0930- 1230
Singapore Mathematical SocietySingapore Mathematical Olympiad (SMO) 2006
(Senior Section, Special Round)
Important:Attempt as many questions as you can.No calculators are allowed.Show all the steps in your working.Each question carries 10 marks.
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1. Let a, d be integers such that a, a + d, a + 2d are all prime numbers larger than 3.Prove that d is a multiple of 6.
2. Let ABCD be a cyclic quadrilateral, let the angle bisectors at A and B meet at E,and let the line through E parallel to side CD intersect AD at Land BC at M.Prove that LA +MB =LM.
3. Two circles are tangent to each other internally at a point T. Let the chord AB ofthe larger circle be tangent to the smaller circle at a point P. Prove that the lineTP bisects LAT B.
4. You have a large number of congruent equilateral triangular tiles on a table andyou want to fit n of them together to make a convex equiangular hexagon (i.e., onewhose interior angles are 120). Obviously, n cannot be any positive integer. Thefirst three feasible n are 6, 10 and 13. Determine if 19 and 20 are feasible.
5. It is claimed that the numberN = 526315789473684210
is a persistent number, that is, if multiplied by any positive integer the result-ing number always contains the ten digits 0,1, ... ,9 in some order with possiblerepetitions.
(a) Prove or disprove the claim.(b) Are there any persistent numbers smaller than the above number?
