# International Mathematical Olympiad 1988 by pptfiles

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```									                                         International Mathematical Olympiad 1989
Hong Kong Preliminary Selection Contest

Section A

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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 x3x 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)
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9.   Let n be a given positive integer. If x and y are positive integers such that xy  nx  ny , find the least and
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greatest possible values of x.                                                                                      (2 marks)
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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 ARCR 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

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