1. We understand a palindrome as a natural number that reads the same forwards and backwards, for example, 16 261. Find the largest four-digit palindrome whose square is also a palindrome.
SOLUTION. Each four-digit palindrome $p=\overline{a b b a}$ can be written in the form

$$
p=a \cdot 1001+b \cdot 110
$$

where $a \in\{1,2, \ldots, 9\}$ and $b \in\{0,1,2, \ldots, 9\}$. Then the square of the number $\overline{a b b a}$ has the form

$$
\begin{aligned}
p^{2}= & a^{2} \cdot 1002001+2 a b \cdot 110110+b^{2} \cdot 12100= \\
= & a^{2} \cdot 10^{6}+2 a b \cdot 10^{5}+\left(b^{2}+2 a b\right) \cdot 10^{4}+ \\
& +\left(2 a^{2}+2 b^{2}\right) \cdot 10^{3}+\left(b^{2}+2 a b\right) \cdot 10^{2}+2 a b \cdot 10^{1}+a^{2} .
\end{aligned}
$$

The last digit of the number $p^{2}$ is thus the same as the last digit of the number $a^{2}$.

For $a \geqq 4$, the number $p^{2}$ is necessarily eight-digit. Its first digit is equal to one of the values $c, c+1, c+2$, where $c$ is the first digit of the two-digit number $a^{2}$. (The maximum carry from a lower order is the number 2.) However, if the given number is a palindrome again, its first and last digit are the same. By comparing the first and last digit of the numbers 16, 25, 36, 49, 64, 81, we see that none of them is of the form $\overline{c(c+2)}, \overline{c(c+1)}$ or $\overline{c c}$.

If $a=3$ and $b \geqq 2$, the number $p^{2}$ is again eight-digit, its last digit is 9 and the first is 1, so it is not a palindrome.

In all other cases, the number $p^{2}$ is seven-digit. Since $a^{2}$ is only one-digit and the notation of the number $p^{2}$ is symmetric, it is necessary that all three values $2 a b, 2 a b+b^{2}$, $2 a^{2}+2 b^{2}$ are less than 10 so that there is no carry to a higher order. Let's discuss three cases:

- $a=3$ : the inequality $2 \cdot 3^{2}+2 b^{2}<10$ is not satisfied by any $b$,
- $a=2$ : the inequality $2 \cdot 2^{2}+2 b^{2}<10$ is satisfied only by $b=0$,
- $a=1$ : the inequality $2 \cdot 1^{2}+2 b^{2}<10$ is satisfied only by $b=0, b=1$, 2002 .

Conclusion: The largest four-digit palindrome satisfying the conditions of the problem is the number