# Background Blur Rule: 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 2 of the school stage of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group IV), grades 9–11. Answer and official solution.*

**Answer:** $\frac{3}{4}$

**Solution.**

There are 36 equally likely pairs $(x, y)$ in total.

Petya's answer and the smartphone's coincide when both decide that there will be blur (“yes”) or both decide that there will be no blur (“no”).

**1) $x \ge 4$ (Petya says “yes”).** The smartphone blurs if $x + y \ge 7$:

$$
\begin{aligned}
x = 4 &: \ y \ge 3 \Rightarrow 4 \text{ options};\\
x = 5 &: \ y \ge 2 \Rightarrow 5 \text{ options};\\
x = 6 &: \ y \ge 1 \Rightarrow 6 \text{ options}.
\end{aligned}
$$

In total, $4 + 5 + 6 = 15$ coincidences.

**2) $x \le 3$ (Petya says “no”).** The smartphone does not blur if $x + y \le 6$:

$$
\begin{aligned}
x = 1 &: \ y \le 5 \Rightarrow 5 \text{ options};\\
x = 2 &: \ y \le 4 \Rightarrow 4 \text{ options};\\
x = 3 &: \ y \le 3 \Rightarrow 3 \text{ options}.
\end{aligned}
$$

In total, $5 + 4 + 3 = 12$ coincidences.

Altogether there are $15 + 12 = 27$ coincidences out of 36; therefore,

$$P(\text{coincidence}) = \frac{27}{36} = \frac{3}{4}.$$