The 11th Asian Pacific Mathematical Olympiad
1999年第11届亚太地区数学奥林匹克数学 
 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.
 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} + a_{2}/2 + a_{3}/3 + a_{n}/n ≥ a_{n }for each positive integer n.
 Let ○1 and ○2 be two circles intersecting at P and Q. The common tangent, closer to P, of ○1 and ○2 touches ○1 at A and ○2 at B. The tangent of ○1 at P meets ○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.
 Determine all pairs (a , b) of integers with the property that the numbers a^{2} + 4b and b^{2} + 4a are both perfect squares.
 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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