# Spam Filter Metrics: 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 I), grades 9–11. Answer and official solution.*

**Answer:** $P = 200$, $TP \in [1, 200]$

**Solution.**

From Accuracy $= 0.8$ with $N = 1000$ we obtain

$$TP + TN = 0.8 \cdot 1000 = 800 \ \Rightarrow\ FP + FN = 200.$$

Since $\text{Precision} = 0.5 = \dfrac{TP}{TP + FP}$, we have $TP = FP$ (and $TP + FP > 0$, hence $TP \ge 1$).

Then

$$FN = 200 - FP = 200 - TP, \qquad P = TP + FN = TP + (200 - TP) = 200,$$

that is, $P$ is fixed and equals 200.

Constraints on $TP$: $FN = 200 - TP \ge 0 \Rightarrow TP \le 200$; also $TP \ge 1$ from the condition $TP + FP > 0$. Therefore,

$$TP \in [1, 200].$$