8.2. Biathletes Arina and Lisa start on the same distance. Their starting numbers are two-digit numbers with the following peculiarity: if the sum of the digits of the starting number is added to the square of the difference of the digits of the number, the result is this number. Find the starting numbers of Arina and Lisa. (7 points)

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# Solution.

Let 10a+b be the desired number. According to the condition $\mathrm{a}+\mathrm{b}+(\mathrm{a}-\mathrm{z})^{2}=10 \mathrm{a}+\mathrm{b}$ or $(a-в)^{2}=9 a$, from which it follows that $a-$ is a perfect square.

By checking the three cases ( $a=1, \mathrm{a}=4$ and $\mathrm{a}=9$ ), we get two solutions: 14 and 90.

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