IMO 1977 LL USA53

Origin: USA

Problem

Find all pairs of integers a and b for which 7a + 14b = 5a2 + 5ab + 5b2.

Solution

The discriminant of the given equation considered as a quadratic equation in b is 196−75a2. Thus 75a2 \leq196 and hence −1 \leqa \leq1. Now the integer solutions of the given equation are easily found: (−1, 3), (0, 0), (1, 2).