1. (2 points) For a natural number $x$, five statements are made:

$$
3 x>91
$$

$$
\begin{aligned}
& x37 \\
& 2 x \geq 21 \\
& x>7
\end{aligned}
$$

It is known that only three of them are true, and two are false. Find $x$.
Solution. Let's transform the original system of inequalities

$$
\begin{aligned}
& x>\frac{91}{3} \\
& x \geq \frac{21}{2} \\
& x>\frac{37}{4} \\
& x>7 \\
& x<120
\end{aligned}
$$

The first inequality does not hold, since otherwise, four inequalities would be satisfied immediately. Therefore, $x \leq \frac{91}{3}$, and the fifth inequality is satisfied.

The second inequality also does not hold, since otherwise, four inequalities would be satisfied: the second, third, fourth, and fifth. Therefore, $x<\frac{21}{2}$.

Now it is immediately clear that the desired number $x=10$.

Answer: $x=10$