IMO 1967 LL SWE49

Let n and k be positive integers such that 1 \leqn \leqN + 1,

imolonglistmathematicsolympiad

IMO 1967 LL SWE49

Origin: SWE

Problem

Let n and k be positive integers such that 1 \leqn \leqN + 1, 1 \leqk \leqN + 1. Show that min n̸=k | sin n −sin k| < 2 N .

Solution

Since sin 1, sin 2, . . . , sin(N +1) \in(−1, 1), two of these N +1 numbers have distance less than 2/N. Therefore | sin n −sin k| < 2/N for some integers 1 \leqk, n \leqN + 1, n ̸= k.