# Photo Contest: Range of Recall: 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 of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence (region group II), grades 9–11. Answer and official solution.*

**Answer:** $\text{Recall} \in \left[\dfrac{2}{425}, \dfrac{12}{17}\right]$.

**Solution.**

From $\dfrac{TP}{TP + FP} = 0.8$ we obtain $FP = \frac{1}{4} TP$, that is,

$$TP = 4k, \quad FP = k, \quad k \in \mathbb{Z},\ k \ge 1.$$

Class constraints: $TP \le 850 \Rightarrow k \le 212$ and $FP \le 150 \Rightarrow k \le 150$, so all $k = 1, 2, \ldots, 150$ are admissible. For each such $k$ we can take

$$FN = 850 - 4k \ge 0, \qquad TN = 150 - k \ge 0,$$

that is, such a configuration is achievable.

Then

$$\text{Recall} = \frac{TP}{P} = \frac{4k}{850}.$$

The minimum is at $k = 1$: $TP = 4, FP = 1, FN = 846, TN = 149$, which gives $\dfrac{4}{850} = \dfrac{2}{425}$. The maximum is at $k = 150$: $TP = 600, FP = 150, FN = 250, TN = 0$, which gives $\dfrac{600}{850} = \dfrac{12}{17}$.

Result: $\text{Recall} \in \left[\dfrac{2}{425}, \dfrac{12}{17}\right]$.