2. In a right-angled triangle, the lengths of the leg $a$ and the hypotenuse $c$ are expressed as two-digit integers, and both numbers use the same set of digits. The length of the leg $b$ is a positive rational number. What can the length of the hypotenuse be? (Provide all answers and prove that there are no others.)
Answer: 65.

Solution. Since $a \neq c$, in two-digit numbers, two different digits must be used, and they will stand in different places. Let $a=10 e+d, c=10 d+e$. Then $b=\sqrt{(10 d+e)^{2}-(10 e+d)^{2}}=3 \sqrt{11(d-e)(d+e)}$.

By the condition, $b$ is rational, so it cannot contain square roots. The number 11 is prime, so one of the factors must be divisible by 11. The factor $d-e<11$ (since $d$ and $e$ are digits), therefore, $d+e$ is divisible by 11. But the maximum value of the sum of two different digits is $9+8=17$, so $d+e=11$. At the same time, $d-e$ must be a perfect square. Checking the possible options $9+2,8+3,7+4,6+5$, we see that this condition is only met for $d=6, e=5$. The triangle has sides $33,56,65$.

Comment. A complete and justified solution is provided - 7 points. A generally correct reasoning containing minor gaps or inaccuracies is provided - 6 points. The notation for two-digit numbers is obtained - 1 point. The expression for $b$ through the radical is obtained - another 2 points. The correct answer is found by trial and error, and it is not proven that there are no other solutions - 1 point. The problem is not solved or solved incorrectly - 0 points.