IMO 1967 LL GDR16

Origin: GDR

Problem

Prove the following statement: If r1 and r2 are real numbers whose quotient is irrational, then any real number x can be approximated arbitrarily well by numbers of the form zk1,k2 = k1r1+k2r2, k1, k2 integers; i.e., for every real number x and every positive real number p two integers k1 and k2 can be found such that |x −(k1r1 + k2r2)| < p.

Solution

First let us notice that all the numbers zm1,m2 = m1r1 + m2r2 (m1, m2 \in Z) are distinct, since r1/r2 is irrational. Thus for any n \inN the in- terval [−n(|r1| + |r2|), n(|r1| + |r2|)] contains (2n + 1)2 numbers zm1,m2,

where |m1|, |m2| \leqn. Therefore some two of these (2n + 1)2 numbers, say zm1,m2, zn1,n2, differ by at most 2n(|r1|+|r2|) (2n+1)2−1 = (|r1|+|r2|) 2(n+1) . By taking n large enough we can achieve that zq1,q2 = |zm1,m2 −zn1,n2| \leqp. If now k is the integer such that kzq1,q2 \leqx < (k+1)zq1,q2, then zkq1,kq2 = kzq1,q2 differs from x by at most p, as desired.