5th APMO 1993


A1. A, B, C is a triangle. X, Y, Z lie on the sides BC, CA, AB respectively, so that AYZ and XYZ are equilateral. BY and CZ meet at K. Prove that YZ² = YK.YB.
A2. How many different values are taken by the expression [x] + [2x] + [5x/3] + [3x]+ [4x] for real x in the range 0 ≤ x ≤ 100?
A3. p(x) = (x + a) q(x) is a real polynomial of degree n. The largest absolute value of the coefficients of p(x) is h and the largest absolute value of the coefficients of q(x) is k. Prove that k ≤ hn.
A4. Find all positive integers n for which xⁿ + (x+2)ⁿ + (2-x)ⁿ = 0 has an integral solution.
A5. C is a 1993-gon of lattice points in the plane (not necessarily convex). Each side of C has no lattice points except the two vertices. Prove that at least one side contains a point (x, y) with 2x and 2y both odd integers.

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