# The Line Rotates Again: Official Solution

*English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email sota.ai.community@gmail.com.*

All-Russian School Olympiad in Informatics 2025–2026, Final Stage<br>
“Artificial Intelligence” profile, Tour 1, Moscow, 23 March 2026<br>
Task A

*The official solutions file repeats each statement before its solution; the statement is in the task statement file. The marking criteria below come from the “Marking criteria for the first tour” pages at the end of the same file.*

## Solution

One example: $A = -4$, $b = -4$, $c = 4$, that is, $P(x) = (x - 2)^2 = x^2 - 4x + 4$. Let $L(x, y) = x - 4y$. Then

$$\begin{aligned}
L(1, P(1)) &= L(1, 1) = -3 \leq 0 \implies \text{the point } (1, P(1)) \text{ belongs to the second class;}\\
L(2, P(2)) &= L(2, 0) = 2 > 0 \implies \text{the point } (2, P(2)) \text{ belongs to the first class;}\\
L(3, P(3)) &= L(3, 1) = -1 \leq 0 \implies \text{the point } (3, P(3)) \text{ belongs to the second class.}
\end{aligned}$$

## Marking criteria

A. Correct answer without verification: 45 points.