# Minimise the Loss: 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.*

*Task 4 of the school stage (Moscow) of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence, grades 9–11: official answer and solution (the statement is in a separate file). Original: [sol-ai-9-11-sch-msk-25-26.pdf](https://vos.olimpiada.ru/upload/files/Arhive_tasks/2025-26/school/ai/sol-ai-9-11-sch-msk-25-26.pdf).*

**Answer:** $x^* = 1,\ y^* = -3,\ \mathcal{L}(x^*, y^*) = 8$

**Scoring criterion:** exact match of the answer — 12 points

**Maximum score for the task — 12**

**Solution.**

Let us complete the squares:

$$
\begin{aligned}
\mathcal{L}(x, y) &= (y + x^2 + 2)^2 - (x^2 + 2)^2 + x^4 + 6x^2 - 4x + 14 \\
&= (y + x^2 + 2)^2 + \left(2x^2 - 4x + 10\right) \\
&= (y + x^2 + 2)^2 + 2(x - 1)^2 + 8.
\end{aligned}
$$

The minimum is attained when the squares are equal to zero:

$$
x - 1 = 0 \Rightarrow x^* = 1, \qquad y + x^2 + 2 = 0 \Rightarrow y^* = -1^2 - 2 = -3.
$$

Then $\mathcal{L}(x^*, y^*) = 8$.