Checklist VsOSh AI 2026 School Stage (Moscow), grades 9–11 · Task 1
Clusters on the Number Line
Russian title: Задание 1
Find the moduli k for which given points split into exactly two clusters of pairwise "friendly" points.
The task
For an integer k ≥ 2, two points on the number line are "friends" if the difference of their coordinates is divisible by k. A non-empty set of points is a cluster if any two of its points are friends and no further point can be added while keeping this property.
For which integers k ≥ 2 can the points with coordinates 1, 7, 21, 22, 28, 42, 43, 49, 63 be split into two clusters?
Abridged and translated by SOTA from the official Russian materials. The official statement has the exact rules, and it wins wherever this summary differs.
In English
This task was published in Russian. SOTA translated its 4 files into English on 16 September 2026.
- Task statement Russian original of Task statement
- Official solution Russian original of Official solution
- Full paper (all tasks of the stage) Russian original of Full paper (all tasks of the stage)
- All solutions of the stage Russian original of All solutions of the stage
Read the task statement in English
Clusters on the Number Line
English translation by SOTA – AI Community of the Russian original. Organisers who would like this translation removed can email [email protected].
Task 1 of the school stage (Moscow) of the All-Russian School Olympiad (VsOSh) 2025/26 in artificial intelligence, grades 9–11. Original: tasks-ai-9-11-sch-msk-25-26.pdf.
Let be an integer. We say that two points on the number line are friends if the difference of their coordinates is divisible by . A non-empty set of points is called a cluster if any two points in this set are friends and no point can be added to it so that this property is preserved.
For which integers can the points of the number line with coordinates
be split into two clusters?
Translator's note: the statement paper gives only the total for the olympiad (112 points); the official solutions give this task a maximum of 12 points (exact match of the answer).
Translated by SOTA. The Russian original is the official version and wins wherever the two differ. Translated from the statement and solution PDFs of the Moscow school stage (grades 9–11) on vos.olimpiada.ru. The statement paper gives no per-task points; the 12 points come from the official solutions. If you organise this olympiad and would like the translation removed, email [email protected] and we will take it down.
At a glance
- You get
- Statement only.
- You submit
- The set of suitable k.
- Scoring
- Exact answer (points per task are not stated in the statement file).
- Format
- School stage (Moscow), started 24 October 2025, grades 9–11; individual; online testing system; maximum 112 points for the paper.