Problem 1. Determine the three-digit numbers $\overline{a b c}$ whose square has the hundreds digit $a$, the tens digit $b$, and the units digit $c$.
Solution and grading. Let $n=\overline{a b c}$. We have $n^{2}=M \cdot 1000+n \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ point

It follows that $n(n-1)=M \cdot 1000$, so $n(n-1)$ is divisible by $2^{3}$ and by $5^{3}$. Since the numbers $n$ and $n-1$ are relatively prime, then $n$ or $n-1$ is divisible by $5^{3}$, and $n$ or $n-1$ is divisible by $2^{3}$

.3 points

From the divisibility by $125$, $n$ can be one of the numbers $125,126,250,251,375,376$, $500,501,625,626,750,751,875,876$. In each case, $n$ or $n-1$ must be divisible by 8. By verification, only the solutions 376 and $625 \ldots \ldots . . \ldots . . . . . . . . . . . . . . . . . . . . . . . . . . .$.