IMO 1977 LL POL33

Origin: POL

Problem

A circle K centered at (0, 0) is given. Prove that for every vector (a1, a2) there is a positive integer n such that the circle K translated by the vector n(a1, a2) contains a lattice point (i.e., a point both of whose coordinates are integers).

Solution

Let r be the radius of K and s > \sqrt 2/r an integer. Consider the points Ak(ka1−[ka1], ka2−[ka2]), where k = 0, 1, 2, . . ., s2. Since all these points are in the unit square, two of them, say Ap, Aq, q > p, are in a small square with side 1/s, and consequently ApAq \leq \sqrt 2/s < r. Therefore, for n = q −p, m1 = [qa1] −[pa1] and m2 = [qa2] −[pa2] the distance between the points n(a1, a2) and (m1, m2) is less then r, i.e., the point (m1, m2) is in the circle K + n(a1, a2).