IMO Solutions Guide
The International Mathematical Olympiad is the world championship of school mathematics. Each July, teams of up to six students from over a hundred countries sit two papers on two days, three problems a day, four and a half hours each. Problems 1 and 4 are usually the gentlest and 3 and 6 the hardest. Here, year by year, are short topic labels, a link to the official problems, and our own solutions.
About these pages. The IMO problems are © the International Mathematical Olympiad. We don't copy them: read each one at the official link, on the IMO's official site, which has the problems of every year in many languages. The hints and solutions are our own work, written from scratch. They are not official solutions.
Years
- 202566th IMO · 5 of 6 solutions ready
- 202465th IMO · 5 of 6 solutions ready
- 202364th IMO · 5 of 6 solutions ready
- 202263rd IMO · 6 of 6 solutions ready
- 202162nd IMO · 6 of 6 solutions ready
- 202061st IMO · 6 of 6 solutions ready
- 201960th IMO · 6 of 6 solutions ready
- 201859th IMO · 6 of 6 solutions ready
- 201758th IMO · 6 of 6 solutions ready
- 201657th IMO · 6 of 6 solutions ready
- 198829th IMO · 1 of 1 solution ready
Every problem
| Problem | Topic | Area | Level | Solution |
|---|---|---|---|---|
| 2025 P1 | Covering a triangle of lattice points with sunny lines | Combinatorics | 2 | Ready |
| 2025 P2 | Two crossing circles, an orthocenter and a tangent line | Geometry | 4 | Ready |
| 2025 P3 | Divisibility functions with exponents: the best linear bound | Number theory | 4 | Ready |
| 2025 P4 | Repeatedly summing the three largest proper divisors | Number theory | 3 | Ready |
| 2025 P5 | A game of sum budgets against square-sum budgets | Algebra | 3 | Ready |
| 2025 P6 | Fewest tiles leaving one hole per row and column | Combinatorics | 5 | Coming |
| 2024 P1 | Sums of floors of multiples, divisible by their count | Number theory, Algebra | 1 | Ready |
| 2024 P2 | When a gcd of two exponential expressions settles | Number theory | 2 | Ready |
| 2024 P3 | A self-counting sequence has an eventually periodic half | Combinatorics | 5 | Coming |
| 2024 P4 | Incircle tangents parallel to sides and an angle sum | Geometry | 3 | Ready |
| 2024 P5 | A snail crossing a board of hidden monsters | Combinatorics | 4 | Ready |
| 2024 P6 | Either-or functional equation on the rationals: symmetric sums | Algebra | 5 | Ready |
| 2023 P1 | Composites whose divisors divide sums of the next two | Number theory | 1 | Ready |
| 2023 P2 | Arc midpoint, altitude, and a tangent meeting the bisector | Geometry | 3 | Ready |
| 2023 P3 | Monic polynomial turning one term into a product of next | Number theory, Algebra | 5 | Ready |
| 2023 P4 | Integer values of sum times reciprocal sum grow fast | Algebra | 2 | Ready |
| 2023 P5 | Best downward path through a triangle of red circles | Combinatorics | 4 | Ready |
| 2023 P6 | Equilateral triangle, three apex points, circles sharing two points | Geometry | 5 | Coming |
| 2022 P1 | Moving coin runs to the front: when do they sort? | Combinatorics | 2 | Ready |
| 2022 P2 | Functions where each x has exactly one partner | Algebra | 2 | Ready |
| 2022 P3 | Primes around a circle with quadratic neighbor products | Number theory | 5 | Ready |
| 2022 P4 | Pentagon with matching lengths: four points on a circle | Geometry | 2 | Ready |
| 2022 P5 | A prime power equal to a factorial plus that prime | Number theory | 2 | Ready |
| 2022 P6 | Fewest increasing paths from local minima on a board | Combinatorics | 5 | Ready |
| 2021 P1 | Split consecutive cards: a square sum is unavoidable | Combinatorics, Number theory | 1 | Ready |
| 2021 P2 | Square roots of differences never beat square roots of sums | Algebra | 4 | Ready |
| 2021 P3 | Angle-bisector point, two circumcenters and a concurrence | Geometry | 5 | Ready |
| 2021 P4 | Tangential quadrilateral and a circle through its incenter | Geometry | 3 | Ready |
| 2021 P5 | Swapping neighbors around a circle: one swap straddles | Combinatorics | 3 | Ready |
| 2021 P6 | Subsets with prescribed sums need a large ground set | Combinatorics, Algebra | 4 | Ready |
| 2020 P1 | Angle ratios 1:2:3 and a shared circumcenter | Geometry | 2 | Ready |
| 2020 P2 | Product of a weighted sum and self-powers stays below one | Algebra | 3 | Ready |
| 2020 P3 | Splitting colored weights into two balanced piles | Combinatorics | 3 | Ready |
| 2020 P4 | Two cable car networks forced to share a connection | Combinatorics | 2 | Ready |
| 2020 P5 | Card deck where pair averages are geometric means | Number theory | 3 | Ready |
| 2020 P6 | Separating line far from all points of a spread set | Geometry, Combinatorics | 5 | Ready |
| 2019 P1 | Integer functional equation with doubling and composition | Algebra, Number theory | 1 | Ready |
| 2019 P2 | Four points on a circle through two cevians | Geometry | 3 | Ready |
| 2019 P3 | Trading friendships until everyone has at most one | Combinatorics | 5 | Ready |
| 2019 P4 | A factorial equal to a product of power-of-two gaps | Number theory | 2 | Ready |
| 2019 P5 | Flipping the coin counted by the heads: average length | Combinatorics | 3 | Ready |
| 2019 P6 | Incircle chords and two circles meeting on a perpendicular | Geometry | 5 | Ready |
| 2018 P1 | Equal tangents from a vertex give a parallel chord | Geometry | 2 | Ready |
| 2018 P2 | Cyclic sequence where each term is product plus one | Algebra | 2 | Ready |
| 2018 P3 | Triangle of absolute differences using each number once | Combinatorics | 4 | Ready |
| 2018 P4 | Stone placing game avoiding one knight-move distance | Combinatorics | 2 | Ready |
| 2018 P5 | Integer cyclic ratio sums force an eventually constant sequence | Number theory | 3 | Ready |
| 2018 P6 | Quadrilateral with balanced side products and an inner point | Geometry | 5 | Ready |
| 2017 P1 | Square root or add three: when does it cycle? | Number theory | 1 | Ready |
| 2017 P2 | Functional equation mixing a composition, a sum and a product | Algebra | 3 | Ready |
| 2017 P3 | Chasing an invisible runner with a noisy tracker | Combinatorics, Geometry | 5 | Ready |
| 2017 P4 | Tangent line from a midpoint construction on a circle | Geometry | 2 | Ready |
| 2017 P5 | Thinning a row so height-ranked pairs stand together | Combinatorics | 3 | Ready |
| 2017 P6 | Homogeneous polynomial equal to one on chosen lattice points | Number theory, Algebra | 4 | Ready |
| 2016 P1 | Three isosceles triangles with equal base angles; a hidden mirror | Geometry | 2 | Ready |
| 2016 P2 | Balanced letters on rows, columns and diagonals of a grid | Combinatorics | 3 | Ready |
| 2016 P3 | Lattice polygon on a circle: area inherits side divisibility | Number theory, Geometry | 5 | Ready |
| 2016 P4 | Shared prime factors among consecutive values of a quadratic | Number theory | 2 | Ready |
| 2016 P5 | Erasing factors so a product equation has no roots | Algebra | 4 | Ready |
| 2016 P6 | Frogs hopping along pairwise crossing segments without collisions | Combinatorics, Geometry | 4 | Ready |
| 1988 P6 | Vieta jumping: a divisibility quotient is a perfect square | Number theory, Algebra | 4 | Ready |