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International Mathematical Olympiad 1989 Hong Kong Preliminary Selection Contest Section A 1 1 1. The function f (x) is defined for all real x. If f (a b) f (ab) for all a, b and f ( ) , compute 2 2 f (1988) . (1 mark) 2. Solve x3x 8 x7 , where x > 0. (1 mark) 3. In an acute triangle, the length of the segment which connects the feet of two of the altitudes is 24, and the mid-point of this segment is M. The length of the side of the triangle not intersected by the segment is 26, and the mid-point of this side is N. Find the length of MN. (1 mark) x 4. How many real roots does the equation sin x have? (1 mark) 1988 5. Find the prime numbers p and q if it is known that the equation x 4 px3 q 0 has an integral root. (1 mark) 6. Let b be an integer greater than 1. For any positive integer n, let Sb (n) be the sum of the digits in the base b representation of n. Let m be a fixed positive integer and r an integer such that 1 r bm . If Sb (r ) x , find (a) Sb (b m 1 r ) , (b) Sb [r (b m 1)] . (2 marks) 7. Let [x] denote the greatest integer not exceeding x. Find the non-integral solutions of x 2 19[ x] 88 0 . (2 marks) 8. Express the area of the inscribed convex octagon in the form r s t , where r, s and t are positive integers. 2 3 2 (2 marks) 3 2 9. Let n be a given positive integer. If x and y are positive integers such that xy nx ny , find the least and 3 2 greatest possible values of x. (2 marks) 3 1 10. Consider the number k k 2 1 where k is a fixed positive integer greater than 1. For n = 1, 2, 3, …, let I n be the integral part of n . Express I 2 n in terms of I n . (2 marks) 11. In the figure, chord AD is bisected by chord BC. The radius of D the circle is 5 and BC = 6. Suppose that AD is the only chord starting at A which is bisected by BC. Find sin AOB, where O B C is the centre of the circle. (Leave the answer as a fraction.) A O (2 marks) Section B Each question carries equal marks. 1. The sides a, b, c of a triangle ABC satisfy a b c or a b c . D and E are the mid-points of AB and BC respectively. The bisectors of BAE and BCD meet at R. Denote the centroid of ABC by G. (a) Prove that 2b2 c 2 a 2 if and only if BDGE is a cyclic quadrilateral. (b) Prove that ARCR if and only if 2b2 c 2 a 2 . (c) Prove that if 2b2 c 2 a 2 , then R lies on the median from B. (d) Is the converse of (c) true? n 3 if n 2000 2. For any integer n, f (n) is defined by f (n) f ( f (n 5)) if n 2000 (a) Find f (1988) . (b) Find all positive integers n such that f (n) 1997 . 2 3. On the circumference of a circle n points are taken. The points are joined pairwise by line segments, dividing the circle into the largest possible number of regions. Denote this number of regions by An . The first few cases are shown below: No. of points (n) Max. no. of regions ( An ) 1 2 2 2 1 3 3 4 2 4 1 4 2 3 4 8 6 5 7 8 1 5 2 3 4 10 8 9 5 16 6 7 14 15 11 13 12 16 (a) Find A8 . (b) Find An in terms of n. END OF PAPER 3