IMC Problem Lab

All 354 problems of the International Mathematics Competition, 1994–2026 — with solutions, and with difficulty measured from what contestants actually scored.

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IMC 2026 →the latest edition · 447 contestants · 10 problems

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Problem catalog →

Filter all 354 problems by topic, year and difficulty; open any and read the solutions.

Collections →

Hand-picked cuts of the archive: Short & very hard, Felled the strong, Natural 6/6 — 10 lists in all.

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Gather problems into a printable sheet; share it as a link.

A problem to try

From the Natural 6/6 collection — recent problems where the question feels inevitable.

IMC 2025 · Day 1 · P1easy

Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\deg(P) \ge 2$ . For every $x \in \mathbb{R}$ , let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point $(x, P(x))$ .

(a) Suppose that the degree of $P$ is odd. Show that $\bigcup\limits_{x \in \mathbb{R}} \ell_x = \mathbb{R}^2$ .

(b) Does there exist a polynomial of even degree for which the above equality still holds?

solved by 82% of the field cohort · avg 8.4/10 · proposed by Mike Daas

IMC 2024 · Day 1 · P3medium

For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0, 1\}$ , such that $A^2$ is the matrix of all ones?

solved by 40% of the field cohort · avg 5.3/10 · proposed by Alex Avdiushenko

Every edition

Dashed years published no individual results — problems and solutions are still there.

Browse by topic

linear algebra · 82 real analysis · 78 combinatorics · 59 inequalities · 54 number theory · 52 sequences & series · 47 polynomials · 47 geometry · 28 integration · 24 group theory · 15 functional equations · 14 complex analysis · 11 probability · 8 set theory & logic · 7 topology & metric spaces · 5 games & processes · 5 abstract algebra · 5

The archive at a glance

Every problem of every edition, labelled by the share of the field that solved it. Each chip links to its problem; on a phone the table scrolls sideways.

The same difficulty bands as everywhere on the site; “—” means the year published no usable per-problem results. Editions 1994–2008 ran six problems a day and are left off this grid — find them in the year list above. Drawn to the dark cells? Hall of pain: the least-solved problems →