MathDB

For every real number

x

, let

||x||

denote the distance between

x

and the nearest integer. Prove that for every pair

(a, b)

of positive integers there exist an odd prime

p

and a positive integer

k

satisfying

\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.

Proposed by Geza Kos, Hungary

N8