5. In a row, the squares of the first 2022 natural numbers are written: $1, 4, 9, \ldots, 4088484$. For each written number, except the first and the last, the arithmetic mean of its left and right neighbors was calculated and written below it (for example, under the number 4, $\left.\frac{1+9}{2}=5\right)$ was written. For the resulting string of 2020 numbers, the same operation was performed. This process continued until a string with only two numbers remained. What are these two numbers?
Solution. Let's look at an arbitrary number in the row $x^{2}$. Under it, it will be written

$$
\frac{(x-1)^{2}+(x+1)^{2}}{2}=\frac{x^{2}-2 x+1+x^{2}+2 x+1}{2}=\frac{2 x^{2}+2}{2}=x^{2}+1
$$

Thus, each time the number increases by one. Initially, there are 2022 numbers, and each time their count decreases by 2, so the operation will be performed 1010 times, and two central numbers will remain, increased by 1010: $1011^{2}+1010=1023131$ and $1012^{2}+1010=$ 1025154.

Criteria. Proven in general that the number increases by 1 each time - 3 points. Solved correctly without a proper proof of the increase by 1 - 4 points. Error in counting the remaining numbers -2 points.