The school maths olympiad in Uzbekistan has four stages: school, district (city), regional and national. The district stage is a test only: 30 tasks (20 closed + 10 open) in 90 minutes. The regional stage has a written paper (5 problems, 180 minutes, 50 points) plus the same kind of test (50 points). Preparation rests on number theory, algebra, combinatorics, probability and plane geometry, and on methods for non-standard problems: parity, the pigeonhole principle, invariants and remainders. Below are the stages, the scoring, the methods and worked examples.
Olympiad stages
The months come from a restatement of the Cabinet of Ministers resolution on olympiads. Check each year's exact dates in the ministry's announcements (gov.uz/edu).
| Stage | Usually when | Format (maths) | Who takes part |
|---|---|---|---|
| I. School | September | Set by the school, no single format | Pupils |
| II. District (city) | October | Test only: 30 tasks (20 closed + 10 open), 90 minutes | School-stage winners, each grade separately (9, 10, 11) |
| III. Regional | November-December | Written paper (5 problems, 180 minutes, 50 points) + test (30 tasks, 90 minutes, 50 points) | District winners (3 per subject and grade) |
| IV. National | March-May | Written paper (problems, proofs) | Regional winners |
The test part: how the points are split
| Tasks | Type | Per task | Total |
|---|---|---|---|
| 1-10 | Closed (choose an answer), simple | 0.9 | 9 |
| 11-20 | Closed, medium | 1.5 | 15 |
| 21-30 | Open (the answer is a number), hard | 2.6 | 26 |
| Total: 30 | 90 minutes | 50 |
- This split is from the 2024 regional-stage test criteria; no separate table has been published for the district stage.
- On a tie, tasks 21-30 are compared first, then 11-20; if still level, the jury sets 3 extra open tasks.
- Calculators, phones and gadgets are banned; pens, rulers, erasers and compasses are allowed.
- From 2026 the "76 points to win" rule is gone: places 1-3 in each subject and grade go by ranking.
Which topics to study
In grades 10-11, topics such as logarithms and the largest and smallest value of a function are added. At the olympiad each grade sits its own paper.
| Area | Key topics |
|---|---|
| Number theory | Divisors, GCD and LCM, primes, remainders, last digits, integer equations |
| Algebra | Identities, equations and inequalities, Vieta's theorem, the mean inequality, sequences |
| Combinatorics | Counting rules, permutations and combinations, the pigeonhole principle, basic graphs |
| Probability | Classical probability, counting favourable outcomes |
| Plane geometry | Triangles, similarity, circles and angles, areas, coordinates |
8 problem-solving methods
- Parity: the product of two consecutive numbers is always even; the parity of a sum sometimes solves a problem in one line.
- The pigeonhole principle: put n + 1 objects into n boxes and some box holds at least 2.
- Remainders (modular arithmetic): last digits, divisibility, "can this number be a square?" questions.
- Invariants: find a quantity that does not change during the process, such as the parity of a sum.
- Small cases: work out n = 1, 2, 3, 4, spot the pattern, then prove it.
- Working backwards: go from the result back, especially in "10 were left at the end" problems.
- The extremal principle: take the largest or smallest element and reason from it.
- Casework: split the problem into a few simple cases and solve each one.
Three worked examples
- What is the last digit of 7 to the power 2026? Powers of 7 end in 7, 9, 3, 1 in turn (period 4). 2026 divided by 4 leaves 2, so the last digit is 9.
- Prove that n² + n is even for every integer n. n² + n = n(n + 1), the product of two consecutive numbers, one of which must be even.
- A class has 13 pupils. Prove that at least two were born in the same month. 12 months are 12 "boxes" and 13 pupils are 13 "objects": by the pigeonhole principle some month has at least 2 pupils.
A study plan
- 4-5 days a week, 1-1.5 hours a day: one topic a day (say, remainders) and 5-8 problems.
- Think about a problem for 20-30 minutes; if it does not give, read the solution and solve it yourself the next day without looking.
- Once a week, take a 30-task test in 90 minutes with no calculator.
- Split your test time: 15-20 minutes on 1-10, 25-30 on 11-20 and the rest on 21-30, which carry the most points (26).
- If you reach the regional stage, practise the written paper on its own: 5 problems, 180 minutes, each written up with a full proof.
Tips for the written paper
- Under the 2024 regional criteria the written problems were worth 7, 7, 10, 10 and 16 points - harder problems pay more.
- The logical chain is marked, not just the answer: write why each step holds.
- Even without a full solution, write up what you have proved (a special case, a lemma).
- Draw neatly with a compass and ruler - in geometry it cuts down mistakes.
Frequently asked questions
- Is there a written paper at the district stage? No, the district stage is a test only.
- How do I write the answer to an open task? As a number: an integer, a decimal or a fraction.
- Where can I find official samples? olimpiada.uzedu.uz has official demo tests.
- What does Trylab offer? Maths tests in the district-stage format: 30 tasks, 90 minutes, 0.9 / 1.5 / 2.6 points, a score out of 50 and a step-by-step solution for every problem (in Uzbek). The written paper (proof problems) is not simulated.
