IMO 1969 LL FRA22

Origin: FRA

Problem

Let \alpha(n) be the number of pairs (x, y) of integers such that x + y = n, 0 \leqy \leqx, and let \beta(n) be the number of triples (x, y, z) such that x + y + z = n and 0 \leqz \leqy \leqx. Find a simple relation between \alpha(n) and the integer part of the number n+2 and the relation among \beta(n), \beta(n −3) and \alpha(n). Then evaluate \beta(n) as a function of the residue of n modulo 6. What can be said about \beta(n) and 1 + n(n+6) ? And what about (n+3)2 ? Find the number of triples (x, y, z) with the property x + y + z \leqn, 0 \leqz \leqy \leqx as a function of the residue of n modulo 6. What can be said about the relation between this number and the number (n+6)(2n2+9n+12) ?