# Maths in a Chatbot: Official 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 1 of the municipal stage (Moscow) of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence, grades 9–11: official answer and solution (the statement is in a separate file). Original: [ans-ai-9-11-mun-msk-25-26.pdf](https://vos.olimpiada.ru/upload/files/Arhive_tasks/2025-26/mun/ai/ans-ai-9-11-mun-msk-25-26.pdf).*

**Answer:** 15

**Scoring criterion:** exact match of the answer — 100 points

**Maximum score for the task — 100**

**Solution.** From the statement we obtain $a^b - ab = 110$. Hence 110 is divisible by $a$. This leaves the options $a = 1, 2, 5, 10, 11, 22, 55, 110$.

$a = 1$: $1 - b = 110$ — no solutions.

$a = 2$: $2^{b-1} - b = 55$. The left-hand side is less than 55 for $b \le 6$ and greater than 55 for $b \ge 7$.

$a = 5$: $5^{b-1} - b = 22$. The left-hand side is less than 22 for $b \le 2$, equal to 22 for $b = 3$, and greater than 22 for $b \ge 4$.

$a = 10$: $10^{b-1} - b = 11$. The left-hand side is greater than 11 for $b \ge 3$; $b = 1, 2$ do not work. The cases $a = 11, 22, 55, 110$ are dealt with in the same way as the previous case.