2015
Problem - 2205
Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
The answer is $\boxed{\frac{1}{20}}$.
Because the $\sin$ function is strictly increasing in $[-\frac{\pi}{2},\ \frac{\pi}{2}]$ which means it is strictly increasing in $[-1,\ 1]$, therefore the function $\sin\sin$ is strictly increasing in $[-\frac{\pi}{2},\ \frac{\pi}{2}]$. Applying the same reasoning can find that $\sin\sin x$ is strictly decreasing in $[\frac{\pi}{2},\ \frac{3\pi}{2}]$.