IMO 2022 Problems and Solutions PDF | International Mathematical Olympiad

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IMO 2022 at a Glance

The 63rd International Mathematical Olympiad (IMO 2022) was held in the beautiful city of Oslo, Norway, bringing together some of the brightest young mathematicians from around the world. The competition took place from July 6 to July 16, 2022, continuing the long-standing tradition of promoting excellence in mathematical problem-solving and international friendship.

A remarkable 104 countries and regions participated in the event, with 589 talented contestants competing for gold, silver, and bronze medals. Over two examination days, each participant attempted six challenging problems covering algebra, geometry, number theory, and combinatorics. Every problem was worth 7 points, making the maximum possible score 42 points.

IMO 2022 was widely praised for its elegant and creative problem set, encouraging students to think beyond standard techniques and develop innovative mathematical ideas. The competition once again demonstrated why the International Mathematical Olympiad is regarded as the world’s most prestigious mathematics contest for high school students.

Whether you are preparing for future Olympiads, studying advanced mathematics, or simply enjoy solving challenging problems, the IMO 2022 Problems and Solutions provide an excellent opportunity to explore beautiful mathematical concepts and strengthen your problem-solving skills.

What You’ll Find on This Page

✔ Complete IMO 2022 problem statements

✔ Detailed step-by-step solutions

✔ Downloadable PDF for offline practice

✔ Difficulty analysis for each problem

✔ Mathematical concepts used in the solutions

✔ Tips for approaching Olympiad-level questions

About IMO 2022

The 63rd International Mathematical Olympiad welcomed hundreds of contestants representing countries from every continent. Each participant attempted six problems over two examination days, with each problem worth 7 points, making a maximum possible score of 42 points.

The paper maintained the traditional IMO structure:

  • Problem 1: Accessible but requires clever observation.
  • Problem 2: Medium difficulty with elegant reasoning.
  • Problem 3: The hardest problem of Day 1.
  • Problem 4: Usually approachable with careful analysis.
  • Problem 5: Advanced problem demanding deep insight.
  • Problem 6: The final challenge, often requiring originality and persistence

Why Study IMO 2022 Problems?

The IMO 2022 problems demonstrate many important Olympiad techniques, including

  • Invariant and monovariant arguments
  • Extremal principles
  • Geometric transformations
  • Number theoretic constructions
  • Functional thinking
  • Combinatorial counting strategies

IMO 2022 Problem List

Problem 1

An elegant introductory problem that combines observation with algebraic reasoning. Although it appears straightforward, finding the correct invariant is the key to a complete solution.

Problem 1

The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k\le 2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k^{th}$ coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n = 4$ and $k = 4$, the process starting from the ordering AABBBABA would be

AABBBABA → BBBAAABA → AAABBBBA → BBBBAAAA → BBBBAAAA → …

Find all pairs $(n, k)$ with $1 \le k \le 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

Problem 2

This problem requires careful logical deductions and rewards systematic thinking rather than lengthy calculations.

Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that for each $x \in \mathbb{R}^+$, there is exactly one $y \in \mathbb{R}^+$ satisfying

image

Problem 3

One of the most challenging questions of Day 1, demanding creativity and a deep understanding of Olympiad techniques.

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around a circle such that the product of any two neighbours is of the form $x^2 + x + k$ for some positive integer $x$

Let

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be a set of odd primes with the property that for every distinct B3def578 543a 4816 97ef 428ab3e434a7,

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for some nonnegative integer 3b3d4a2e 4b87 45cf 9632 E9f1a9df40a7.

We must prove that

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Problem 4

A beautifully structured problem where recognizing hidden relationships leads to an elegant proof

Problem 4

Let $ABCDE$ be a convex pentagon such that $BC = DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB = TD$, $TC = TE$ and $\angle ABT = \angle TEA$.

Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P, B, A, Q$ occur on their line in that order. Let line $AE$ intersect lines $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R, E, A, S$ occur on their line in that order. Prove that the points $P, S, Q, R$ lie on a circle.

Problem 5

An advanced problem that tests persistence and the ability to connect multiple mathematical ideas.

Find all triples $(a,b,p)$ of positive integers with $p$ prime and\[a^p = b! + p\]

Problem 6

The final problem of th competition presents a rich mathematical structure and showcases the depth and beauty of Oly

Let $n$ be a positive integer. A Nordic square is an $n \times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share an edge. Every cell that is adjacent only to cells containing larger numbers is called a valley. An uphill path is a sequence of one or more cells such that:

(i) the first cell in the sequence is a valley,

(ii) each subsequent cell in the sequence is adjacent to the previous cell, and

(iii) the numbers written in the cells in the sequence are in increasing order.

Find, as a function of $n$, the smallest possible total number of uphill paths in a Nordic square.

Top 10 Team Rankings

RankCountryScore
🥇 1China252/252
🥈 2South Korea208
🥉 3United States207
4Vietnam196
5Romania194
6Thailand193
7Germany192
8Japan190
8Iran190
10Israel189

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