Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it Soviet Union

Problem:
Let $f(x) = a\cos(x + 1) + b\cos(x + 2) + c\cos(x + 3)$ , where $a$ , $b$ , $c$ are real. Given that $f(x)$ has at least two zeros in the interval $(0,\pi)$ , find all its real zeros.

Solution

Solution:
Answer: $f(x)$ must be identically zero.

We have $f(x) = (a\cos 1 + b\cos 2 + c\cos 3)\cos x - (a\sin 1 + b\sin 2 + c\sin 3)\sin x$ . This can be written as $d\cos(x + \theta)$ for some $d$ , $\theta$ . But if $d \neq 0$ , then this has only one zero in the interval $(0,\pi)$ . Hence $d = 0$ .

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