13th APMO 2001


A1. If n is a positive integer, let d be the number of digits in n (in base 10) and s be the sum of the digits. Let n(k) be the number formed by deleting the last k digits of n. Prove that n = s + 9 n(1) + 9 n(2) + ... + 9 n(d).
A2. Find the largest n so that the number of integers less than or equal to n and divisible by 3 equals the number divisible by 5 or 7 (or both).
A3. Two equal-sized regular n-gons intersect to form a 2n-gon C. Prove that the sum of the sides of C which form part of one n-gon equals half the perimeter of C.
A4. Find all real polynomials p(x) such that x is rational iff p(x) is rational.
A5. What is the largest n for which we can find n + 4 points in the plane, A, B, C, D, X₁, ... , X_n, so that AB is not equal to CD, but for each i the two triangles ABX_i and CDX_i are congruent?

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