Baltic Way 2021 · Shortlist problem

Geometry

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 $\sin(x) = \sin(a+x)$ .

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Topics

Coordinates and vectors

Solutions

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. $\square$