Version : English
First Day - Taipei - July 15, 1998

### Problem 1

In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. Suppose that the point P, where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove that ABCD is a cyclic quadrilateral if and only if the triangles ABP and CDP have equal areas.

### Problem 2

In a competition, there are a contestants and b judges, where b ³ 3 is an odd integer. Each judge rates each contestant as either ``pass'' or ``fail''. Suppose k is a number such that, for any two judges, their ratings coincide for at most k contestants. Prove that
 k  a ³ b-1  2b .

### Problem 3

For any positive integer n, let d(n) denote the number of positive divisors of n     (including 1 and n itself).

Determine all positive integers k such that

 d(n2)  d(n) = k

for some n.

Time Allowed : 4[1/2] hours.

Each problem is worth 7 points.

Version : English
Second Day - Taipei - July 16, 1998

### Problem 4

Determine all pairs (a,b) of positive integers such that ab2+b+7 divides a2b+a+b.

### Problem 5

Let I be the incentre of triangle ABC. Let the incircle of ABC touch the sides BC, CA and AB at K, L and M, respectively. The line through B parallel to MK meets the lines LM and LK at R and S, respectively. Prove that ÐRIS is acute.

### Problem 6

Consider all functions f from the set N of all positive integers into itself satisfying
 f( t2f( s) ) = s( f( t) )2,

for all s and t in N. Determine the least possible value of f(1998).

Time Allowed : 4[1/2] hours.

Each problem is worth 7 points.