MathDB

Let

m

and

n

be two distinct positive integers. Prove that there exists a real number

x

such that

\frac {1}{3}\le\{xn\}\le\frac {2}{3}

and

\frac {1}{3}\le\{xm\}\le\frac {2}{3}

. Here, for any real number

y

\{y\}

denotes the fractional part of

y

. For example \{3.1415\} \equal{} 0.1415.

Problem Statement