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Problem 860

AMC 12 late, AIME early

Number theory Difficulty 4.6 Prove it Croatian Mathematical Competitions · Croatia

Prove that there is no integer $n \ge 2$ such that
$f(x) = \cos(x\sqrt{1}) + \cos(x\sqrt{2}) + \dots + \cos(x\sqrt{n})$
is a periodic function.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Suppose, on the contrary, that the function $f$ is periodic with the period $T$ for some integer $n \ge 2$ . Hence, $f(T) = f(0) = n$ .
Now we have
$f(T) = \cos(T\sqrt{1}) + \cos(T\sqrt{2}) + \dots + \cos(T\sqrt{n}) = n,$
from which we conclude that $\cos(T\sqrt{1}) = \cos(T\sqrt{2}) = \dots = \cos(T\sqrt{n}) = 1$ . So, $T = 2k\pi$ and $T\sqrt{2} = 2l\pi$ , where $k, l \in \mathbb{N}$ . Hence, $\sqrt{2} = l/k \in \mathbb{Q}$ , which is contradiction. In conclusion, there is no integer $n \ge 2$ such that $f$ is periodic.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.