Balti Tee 2021 · Valikvooru ülesanne

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. $\square$