11^{th} Asian Pacific Mathematical Olympiad

March, 1999

1. Find the smallest positive integer n with the following property: there does not exist an arithmetic progression of 1999 real numbers containing exactly n integers.

2. Let a_{1}, a_{2}, … be a sequence of real numbers satisfying a_{i+j} ≤ a_{i}+a_{j} for all i, j = 1, 2, . . ..

Prove that

a_{1} + \frac{a_{2}}{2} + \frac{a_{3}}{3} + ⋯ + \frac{a_{n}}{n} ≥ a_{n}

for each positive integer n.

3. Let Γ_{1} and Γ_{2} be two circles intersecting at P and Q. The common tangent, closer to P, of Γ_{1} and Γ_{2} touches Γ_{1} at A and Γ_{2} at B. The tangent of Γ_{1} at P meets Γ_{2} at C, which is different from P, and the extension of AP meets BC at R. Prove that the circumcircle of triangle P QR is tangent to BP and BR.

4. Determine all pairs (a, b) of integers with the property that the numbers a^{2} + 4b and b^{2} + 4a are both perfect squares.

5. Let S be a set of 2n + 1 points in the plane such that no three are collinear and no four concyclic. A circle will be called good if it has 3 points of S on its circumference, n − 1 points in its interior and n − 1 points in its exterior. Prove that the number of good circles has the same parity as n.