# Two Positive Examples: 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 5 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:** 7

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

**Maximum score for the task — 12**

**Solution.**

Let the number of negative examples be $m = N - 2$.

The number of 4-element groups:

$$
\#\text{total} = \frac{N(N-1)(N-2)(N-3)}{24}.
$$

Groups with both positive examples: we need to add 2 of the $m$ negative examples,

$$
\#(2^+) = \frac{m(m-1)}{2}.
$$

Groups without positive examples: we take all quadruples of the $m$ negative examples,

$$
\#(0^+) = \frac{m(m-1)(m-2)(m-3)}{24}.
$$

By the statement,

$$
\frac{\#(2^+)}{\#\text{total}} = 2 \cdot \frac{\#(0^+)}{\#\text{total}} \implies \frac{m(m-1)}{2} = 2 \cdot \frac{m(m-1)(m-2)(m-3)}{24}.
$$

Cancelling (for $m \geq 4$):

$$
\frac{1}{2} = \frac{(m-2)(m-3)}{12} \implies (m-2)(m-3) = 6 \implies m = 5.
$$

Hence $N = m + 2 = 7$.