1. Find the minimum value of the expression $\frac{25 x^{2} \sin ^{2} x+16}{x \sin x}$ for $0<x<\pi$.
Answer: 40.

Solution. Let $y = x \sin x$. Then the expression can be written as

$$
\frac{25 y^{2} + 16}{y} = 25 y + \frac{16}{y}.
$$

Note that $y > 0$ (since $x > 0$ and $\sin x > 0$), so we can apply the inequality between the means:

$$
25 y + \frac{16}{y} \geq 2 \sqrt{25 y \cdot \frac{16}{y}} = 40.
$$

Equality holds when $25 y = \frac{16}{y}$, i.e., $y = \frac{4}{5}$, which is a valid value.

Comment. A complete and justified solution is given - 7 points. A generally correct reasoning with minor gaps or inaccuracies - 6 points. The correct answer is obtained, but its minimality is not proven - 2 points. A new variable is introduced without further progress - 1 point. Only special cases or specific examples are considered - 1 point. The problem is not solved or solved incorrectly - 0 points.