Problem 1040

Let, for example, $f(x)=\cos 2 / 3 \pi x+\cos \pi x, g(x)=-\cos \pi x$, then their sum $h(x)$ is $\cos 2 / 3 \pi x$.

The smallest positive period of the function $g(x)$ is $2 \pi: \pi=2$, and the smallest positive period of the function $h(x)$ is $2 \pi: 2 \pi / 3=3$.

We will prove that the smallest positive period of the function $f(x)$ is 6. Indeed, the number 6 is a multiple of 3 and 2, so it is both a period of the function $h(x)$ and a period of the function $g(x)$, which means it is also a period of their difference.

Suppose that a positive number $T$ is a period of the function $f(x)$. Then

$\cos ^{2} / 3 \pi T+\cos \pi T=\cos 0+\cos 0=2 \Leftrightarrow \cos ^{2} / 3 \pi T=\cos \pi T=1 \Leftrightarrow \cos ^{2} / 3 \pi T=\cos \pi T=1 \Leftrightarrow 2 / 3 \pi T=2 \pi k, \pi T=$ $2 \pi n, k, n \in \mathbf{N}$. Therefore,

$T=3 k=2 n$. This means that $T$ is an integer that is a multiple of both 2 and 3, so it is at least 6.

## Answer

There exist.