# Homework or Answer Key

*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. The official answer and solution are in a separate file.*

When programs are taught from examples (this is called machine learning), a certain mistake happens: the correct answers accidentally end up in the training examples. The program then does not learn to reason but simply memorises the answers, and fails on new problems. Schoolchildren are in a similar situation: one can honestly work through a topic, or one can copy from “ready-made homework” answer books (GDZ). By copying, it is easy to get high marks, but there is little knowledge, and this will show in the test.

A pupil has 10 homework assignments and one test. For each homework assignment, he chooses either to solve it on his own (H) or to copy it from the GDZ (G).

Let $K_t$ be the level of knowledge before the $t$-th homework assignment ($t$=1, ..., 10). If the pupil solves assignment $t$ himself, his level of knowledge increases by one; otherwise the level of knowledge does not increase; $K_1 = 0$.<br>
The mark for the $t$-th homework assignment is calculated as follows: if it was solved honestly, then $H_t = 1.5 + 0.3K_t$; if the homework was copied, then $H_t = 4.5$.<br>
The mark for the test depends only on the knowledge after the last homework assignment, $K_{11}$:

$$E = \min\{1 + 0.5K_{11}, 5\}$$

The final average mark: 40% for the homework, 60% for the test:

$$S = 0.4 \frac{H_1 + H_2 + ... + H_{10}}{10} + 0.6E$$

Find the maximum value of $S$. Round the answer to thousandths.<br>
Find the optimal number of honestly completed homework assignments at which $S$ is maximal.

**Scoring criterion:** 6 points for each correct answer. Total — 12 points

**Maximum score for the task — 12**