1. Find all $x$ that satisfy the inequality $n^{2} x^{2}-\left(2 n^{2}+n\right) x+n^{2}+n-6 \leq 0$ for any natural $n$.
Solution. The roots of the quadratic trinomial $n^{2} x^{2}-\left(2 n^{2}+n\right) x+n^{2}+n-6$ are

$$
x_{1}=1-\frac{2}{n}, x_{2}=1+\frac{3}{n}
$$

Factorize the left side of the inequality

$$
n^{2}\left(x-\left(1-\frac{2}{n}\right)\right)\left(x-\left(1+\frac{3}{n}\right)\right) \leq 0
$$

Solving the inequality using the interval method, we get $x \in\left[1-\frac{2}{n} ; 1+\frac{3}{n}\right]$. Only $x=1$ belongs to this interval for any $n$.

Answer: $x=1$