# Homework or Answer Key: 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 3 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:** $S_{\max} = 4.176$, $\quad$ the optimal number of honest homework assignments $h = 8$

**Solution.**

**Idea.** What matters is not which assignments are done honestly but *how many* of them: let there be $h$ honest ones; then the knowledge before the test is $K_{11} = h$.

While the test mark is not yet “5” ($h \le 7$), the exam mark grows by $+0.5$, which gives $+0.6 \cdot 0.5 = +0.3$ in the total $S$. The homework part gets worse when one “4.5” is replaced by an honest assignment ($1.5 + 0.3h$), but by no more than 0.12 in $S$ (the rough upper bound is attained at the lowest knowledge). Hence the total gain is positive — it pays to increase $h$ up to 8.

Once “5” has been reached ($h \ge 8$), the exam mark no longer grows, while honest homework still gives less than 4.5. Hence, from then on $S$ only decreases. So the maximum is at $h = 8$.

**The value of the maximum (substitution into the formula).**

$$S_{\max} = 0.4 \cdot \frac{H_1 + \cdots + H_{10}}{10} + 0.6E = 0.4 \cdot \frac{1.5 + 0.3 \cdot 1 + \cdots + 1.5 + 0.3 \cdot 7 + 4.5 + 4.5}{10} + 0.6 \cdot 5 = 4.176$$

**Answer:** $\boxed{S_{\max} = 4.176, \quad h_{\text{opt}} = 8}$.

*Moral.* You have the opportunity to copy your homework, but what really matters is not marks but knowledge and skills. And these are built up only by honest work.