11th APMO 1999


A1. Find the smallest positive integer n such that no arithmetic progression of 1999 reals contains just n integers.
A2. The real numbers x₁, x₂, x₃, ... satisfy x_i+j ≤ x_i + x_j for all i, j. Prove that x₁ + x₂/2 + ... + x_n/n ≥ x_n.
A3. Two circles touch the line AB at A and B and intersect each other at X and Y with X nearer to the line AB. The tangent to the circle AXY at X meets the circle BXY at W. The ray AX meets BW at Z. Show that BW and BX are tangents to the circle XYZ.
A4. Find all pairs of integers m, n such that m² + 4n and n² + 4m are both squares.
A5. A set of 2n+1 points in the plane has no three collinear and no four concyclic. A circle is said to divide the set if it passes through 3 of the points and has exactly n - 1 points inside it. Show that the number of circles which divide the set is even iff n is even.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.