IMO Problem

An IMO Problem is a mathematical word problem from a International Mathematical Olympiad (IMO).

• Context:
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• Example(s):
  • 1996 IMO Problem 1: Let ABC be a triangle with AB = c, AC = b, and BC = a. Let P be the incenter triangle of ABC, and let r be the inradius of triangle ABC'. Prove that: [math]\displaystyle{ PA + PB + PC = 2r \left( \frac{a}{\sin A} + \frac{b}{\sin B} + \frac{c}{\sin C} \right). }[/math]
  • 2020 IMO Shortlisted Problem 1:
    A sequence is called a $p$-sequence if it can be obtained by starting with a positive integer and then repeatedly adding or subtracting $p$ to or from the previous term. For example, the sequence $1, 4, 1, 4, 1, 4, \ldots$ is a $3$-sequence.

    Prove that there exist infinitely many primes $p$ for which there exist $p$-sequences $pa_0, a_1, a_2, \ldots$ and $pb_0, b_1, b_2, \ldots$ such that $a_n \neq b_n$ for infinitely many $n$, and $b_n \neq a_n$ for infinitely many $n$.

  • 2019 IMO Shortlisted Problem 5:
    On a certain social network, there are 2019 users, some pairs of which are friends. A group of users is called a clique if every pair of users in the group are friends. Prove that there exists a clique containing at most 132 users such that every user in the social network is friends with at least one user in the clique.
  • ...
• See: International Science Olympiad, International Mathematical Olympiad Selection Process.

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References

2023

• (Wikipedia, 2023) ⇒ https://en.wikipedia.org/wiki/International_Mathematical_Olympiad Retrieved:2023-10-23.
  • The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. It is “the most prestigious” mathematical competition in the world. Winning in IMO is widely regarded as the greatest feat for any high school student. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More than 100 countries participate. Each country sends a team of up to six students, plus one team leader, one deputy leader, and observers. The content ranges from extremely difficult algebra and pre-calculus problems to problems in branches of mathematics not conventionally covered in secondary or high school and often not at university level either, such as projective and complex geometry, functional equations, combinatorics, and well-grounded number theory, of which extensive knowledge of theorems is required. Calculus, though allowed in solutions, is never required, as there is a principle that anyone with a basic understanding of mathematics should understand the problems, even if the solutions require a great deal more knowledge. Supporters of this principle claim that this allows more universality and creates an incentive to find elegant, deceptively simple-looking problems which nevertheless require a certain level of ingenuity, often times a great deal of ingenuity to net all points for a given IMO problem. The selection process differs by country, but it often consists of a series of tests which admit fewer students at each progressing test. Awards are given to approximately the top-scoring 50% of the individual contestants. Teams are not officially recognized—all scores are given only to individual contestants, but team scoring is unofficially compared more than individual scores. Contestants must be under the age of 20 and must not be registered at any tertiary institution. Subject to these conditions, an individual may participate any number of times in the IMO.

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