Problem 6. How many solutions does the equation

$$
\arcsin 2x + \arcsin x = \frac{\pi}{3} ?
$$
Solution. The function $f(x)=\arcsin 2 x$ is defined on the interval $[-0.5 ; 0.5]$, and the function $\varphi(x)=\arcsin x$ is defined on the interval $[-1 ; 1]$. Therefore, the left side of equation (1) is defined on the interval $[-0.5 ; 0.5]$. The monotonic and continuous function $y=\arcsin 2 x+\arcsin x$ changes on the interval $[-0.5 ; 0.5]$ from $\left(-\frac{\pi}{2}-\frac{\pi}{6}\right)$ to $\left(\frac{\pi}{2}+\frac{\pi}{6}\right)$, i.e., from $-\frac{2}{3} \pi$ to $\frac{2}{3} \pi$. But $\frac{\pi}{3}<\frac{2}{3} \pi$. Therefore, equation (1) has a unique solution.