Philippine Mathematical Olympiad

Solution:

(ans. 3 .
$\sin x = \log x \Rightarrow x \leq 10.10 < 2 \cdot 2\pi$ means that in $[0, 2\pi]$ there is a complete period of $\sin$ and part of a second period. There is an intersection in the first period, and after the first period, near $\frac{5\pi}{2}$ to the left and to the right, $\sin$ has value $<1$ (and $>0$, of course). Since $2\pi + \frac{\pi}{2} = \frac{5\pi}{2} < 10$, $\sin x$ intersect $\log x$ at two points to the left and right side of $\frac{5\pi}{2}$.)