IMO 1977 LL USS49

Origin: USS

Problem

Find all pairs of integers (p, q) for which all roots of the trino- mials x2 + px + q and x2 + qx + p are integers.

Solution

If one of p, q, say p, is zero, then −q is a perfect square. Conversely, (p, q) = (0, −t2) and (p, q) = (−t2, 0) satisfy the conditions for t \inZ. We now assume that p, q are nonzero. If the trinomial x2 + px + q has two integer roots x1, x2, then |q| = |x1x2| \geq|x1| + |x2| −1 \geq|p| −1. Similarly, if x2 + qx + p has integer roots, then |p| \geq|q| −1 and q2 −4p is a square. Thus we have two cases to investigate: (i) |p| = |q|. Then p2 −4q = p2 \pm 4p is a square, so (p, q) = (4, 4). (ii) |p| = |q|\pm1. The solutions for (p, q) are (t, −1−t) for t \inZ and (5, 6), (6, 5).