THE 1999 ASIAN PACIFIC MATHEMATICAL OLYMPIAD


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} \leq 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} \geq a_n
for each positive integer n.

3. Let \Gamma_1 and \Gamma_2 be two circles intersecting at P and Q. The common tangent, closer to P, of \Gamma_1 and \Gamma_2 touches \Gamma_1 at A and \Gamma_2 at B. The tangent of \Gamma_1 at P meets \Gamma_2 at C, which is different from P, and the extension of AP meets BC at R. Prove that the circumcircle of triangle PQR 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.

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