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International Mathematical Olympiad 1988

VIEWS: 10 PAGES: 3

									                                         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 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)
                                                                                    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 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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