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Problem 2153

National Olympiad second round; IMO P1/P4

Geometry Difficulty 7.8 Prove it Baltic Way Shortlist · Baltic Way · 2021

Let $a$ be a real number. Prove that there exist real numbers $b$ and $c$ such that the inequalities
$\min \{\sin(x), \sin(a+x)\} \le b \sin(x+c) \le \max \{\sin(x), \sin(a+x)\}$
hold for all real numbers $x$ and the equalities hold only if $\sin(x) = \sin(a+x)$ .

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

We recall that two real numbers $r$ and $s$ always satisfy the trigonometric identity $\frac{1}{2}[\sin(r+s) + \sin(r-s)] = \cos(s) \sin(r)$ .
Substituting $r = x + \frac{a}{2}$ and $s = \frac{a}{2}$ , we see that $\cos(\frac{a}{2}) \sin(x + \frac{a}{2})$ is for all real numbers $x$ the arithmetic mean of the numbers $\sin(x + a)$ and $\sin(x)$ . Since the arithmetic mean of two numbers lies in the closed interval bounded by the two numbers, $b = \cos(\frac{a}{2})$ and $c = \frac{a}{2}$ is a suitable choice for $b$ and $c$ . Furthermore, the arithmetic mean of two numbers differs from the two numbers if they are not equal. $\square$

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