Problem 5. Find all two-digit numbers $x$ for which two of the following statements are true, and two are false:

1) $x$ is divisible by 5;
2) $x$ is divisible by 23;
3) $x+7$ is a square of some natural number;
4) $x-10$ is a square of some natural number.
Solution. The sets corresponding to statements 1) - 4) are: For 1) it is the set

$$
A=\{10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95\}
$$

for 2) the set

$$
B=\{23,46,69,92\}
$$

for 3) the set

$$
C=\{18,29,42,57,74,93\} \text { and }
$$

for 4) the set

$$
D=\{11,14,19,26,35,46,59,74,91\}
$$

Then

$$
A \cap B=\varnothing, A \cap C=\varnothing, A \cap D=\{35\}, B \cap C=\varnothing, B \cap D=\{46\}, C \cap D=\{74\}
$$

From $A \cap D=\{35\}$ and $35 \notin B$, and $35 \notin C$ we conclude that 35 is a number for which statements 1) and 4) are true, and statements 2) and 3) are false. From $B \cap D=\{46\}$ and $46 \notin A$, and $46 \notin C$ we conclude that 46 is a number for which statements 2) and 4) are true, and statements 1) and 3) are false. From $C \cap D=\{74\}$ and $74 \notin A$ and $74 \notin B$ we conclude that 74 is a number for which statements 3) and 4) are true, and statements 1) and 2) are false. Therefore, the required two-digit numbers are 35, 46, and 74.

## VIII Department