Singapore Mathematical Society Singapore Mathematical Olympiad (SMO) 2009 (Senior Section) Tuesday, 2 June 2009 0930 — 1200 hrs Important: Answer ALL 35 questions. Enter your answers on the answer sheet provided. For the multiple choice questions, enter your answer on the answer sheet by shading the bubble containing the letter (A, B, C, D or E) corresponding to the correct answer. For the other short questions, write your answer on the answer sheet and shade the appropriate bubble below your answer. No steps are needed to justify your answers. Each question carries 1 mark. No calculators are allowed. PLEASE DO NOT TURN OVER UNTIL YOU ARE TOLD TO DO SOMultiple Choice Questions 1 Suppose that x is a plane and 4 and B are two points on the plane x. If the distance between A and B is 33 cm, how many lines are there in the plane such that the distance between each line and is 7 cm and the distance between each line and B is 26 cm respectively? (A) (B) © (D) (E) Infinitely many Rone Let y= (17 = x)(19 —x)(19 + x)(17 + x), where x is a real number. Find the smallest possible value of y (A) 1296 (B) 1295 (© -1294 () 1293 ©) -1292 If two real numbers a and b are randomly chosen from the interval (0, 1), find the probability that the equation x+b=Ohas real roots. (A) i om = © is (D) t © 4 Ifx and y are real numbers for which |x|+x-+5y=2 and |y|~y+x=7, find the value of x+y. “a 3 (By -l ool (D) 3 («) 5 23Ina triangle ABC, sin A = A) (B) © @) (8) The area of a triangle ABC is 40 em?, Points D, E and Fare on sides AB, BC and CA respectively, as shown in the figure below. If AD area of triangle ABE is equal to the area of quadrilateral DBEF, find the area of triangle AEC in cm’. (A) (B) © @) () Find the value of «) 65 pope 16) Bo 6565 Ww 12 1B 14 15 1h2H3! pleats 2 24! 5 and cos B= +. Find the value of cos C. 2H3H4! om, DB = 5 cm, and the | | |10, There are eight envelopes numbered 1 to 8. Find the number of ways in which 4 identical red buttons and 4 identical blue buttons can be put in the envelopes such that each envelope contains exactly one button, and the sum of the numbers on the envelopes containing the red buttons is more than the sum of the numbers on the envelopes containing the blue buttons. (A) 35 (B) 34 () 32 @) 31 (BE) 62 Determine the number of acute-angled triangles (i.c., all angles are less than 90°) in which all angles (in degrees) are positive integers and the largest angle is three times the smallest angle. (A) 3 (B) 4 © 5 (D) 6 a, Let ABCD be a quadrilateral inscribed in a circle with diameter AC, and let E be the foot of perpendicular from D onto 4B, as shown in the figure below. If AD = DC and the area of quadrilateral ABCD is 24 em’, find the length of DE in cm. D B (Ay 32 (By 2N6 © wT (D) v2 25Short Questions 11, Find the number of positive divisors of (2008? + (3 2008 x 2009) + 1)”. 12. Suppose that a, 6 and c are real numbers greater than 1. Find the value of 1 1 1 —- Te [a 141 I+log,. 13, Find the remainder when (1! x 1) + (2! 2)-+ (3! x 3) + + 286! * 286) is divided by 2009. log... 14, Find the value of (25+10V3)"” +25 ~-10V5) _ 1+ 2009 15. Leta aa Find the value of (a° —503a—500)"". 16. Inthe figure below, ABC's a triangle and D isa point on side BC. Point Eis on side 4B such that DE is the angle bisector of ZADB, and point F is on side AC such that DF is the angle bisector of ZADC. Find the value of 22.82. CF EB DC FA 17. Find the value of (cot25" -1)(eot 24° ~1)(cot23* —1)(cot 22" —1)(cot 21° ~1)(cot20° - 1) 18. Find the number of 2-element subsets {a, 5} of {1, 2,3, ...,99, 100} such that ab + a+ bisa multiple of 7. 26 | |19. 20. 2 22. 23, Let x be a real number such that x* —15x+1=0. Find the value of x* + In the figure below, ABC is a triangle with AB = 10 em and BC = 40 cm. Points D and E lie on side AC and point F on side BC such that EF is parallel to AB and DF is parallel to EB. Given that BE is an angle bisector of ZABC and that AD = 13.5 em, find the length of CD in cm. . Let S= {1, 2, 3, ... , 64, 65}. Determine the number of ordered triples (x, y, 2) such that x,y,2e 5, x 0 such that a+b+¢ = 1. Show that if if x1,02,...,«5 are positive real numbers such that x47r2...25 = 1, then (ax? + bry + c)(ax} + bre +0) +++ ( 5. In an archery competition, there are 30 contestants. The target is divided in two zones. A hit at zone 1 is awarded 10 points while a hit at zone 2 is awarded 5 points. No point is awarded for a miss. Bach contestant shoots 16 arrows. At the end of the competition statistics show that more than 50% of the arrows hit zone 2. The number of arrows that hit zone 1 and miss the target are equal. Prove that there are two contestants with the same score. 44