IMO 1982 LL BRA9

Let n be a natural number, n \geq2, and let \varphi be Euler’s function;

imolonglistmathematicsolympiad

IMO 1982 LL BRA9

Origin: BRA

Problem

Let n be a natural number, n \geq2, and let \varphi be Euler’s function; i.e., \varphi(n) is the number of positive integers not exceeding n and coprime to n. Given any two real numbers \alpha and \beta, 0 \leq\alpha < \beta \leq1, prove that there exists a natural number m such that \alpha < \varphi(m) m < \beta.