art of problem solving imo problems

Evan Chen《陳誼廷》

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Olympiad Problems and Solutions

This page contains problems and solutions to several USA contests, and a few others.

Check the AoPS contest index for even more problems and solutions, including most of the ones below.

Hardness scale #

Here is an index of many problems by my opinions on their difficulty and subject. The difficulties are rated from 0 to 50 in increments of 5, using a scale I devised called MOHS . 1

In 2020, Rustam Turdibaev and Olimjon Olimov, compiled a 336-problem index of recent problems by subject and MOHS rating . In addition, the linked file also contains a hyperlink to each of the corresponding solution threads on Art of Problem-Solving.

This document will probably see a lot of updates. Anyway, I cannot repeat enough the disclaimer that the ratings (and even philosophy) are my own personal opinion, rather than some sort of indisputable truth.

USA Math Olympiad (USAMO) #

Despite being part of the USA team selection process, these are not the “official” solution files, rather my own personal notes. In particular, I tend to be more terse than other sources.

My understanding is that the internal problems and solutions, from the actual USA(J)MO committee, are copyrighted by MAA. To my knowledge they are not published anywhere. The Math Magazine has recently resumed publishing yet another version of the problems and solutions of the olympiad.

Recent statistics for USAMO

Download statistics for 2015-present (PDF) .

Problems and solutions to USAMO

• USAMO 1996 (PDF) (TeX)
• USAMO 1997 (PDF) (TeX)
• USAMO 1998 (PDF) (TeX)
• USAMO 1999 (PDF) (TeX)
• USAMO 2000 (PDF) (TeX)
• USAMO 2001 (PDF) (TeX)
• USAMO 2002 (PDF) (TeX)
• USAMO 2003 (PDF) (TeX)
• USAMO 2004 (PDF) (TeX)
• USAMO 2005 (PDF) (TeX)
• USAMO 2006 (PDF) (TeX)
• USAMO 2007 (PDF) (TeX)
• USAMO 2008 (PDF) (TeX)
• USAMO 2009 (PDF) (TeX)
• USAMO 2010 (PDF) (TeX)
• USAMO 2011 (PDF) (TeX)
• USAMO 2012 (PDF) (TeX)
• USAMO 2013 (PDF) (TeX)
• USAMO 2014 (PDF) (TeX)
• USAMO 2015 (PDF) (TeX)
• USAMO 2016 (PDF) (TeX)
• USAMO 2017 (PDF) (TeX)
• USAMO 2018 (PDF) (TeX)
• USAMO 2019 (PDF) (TeX) (Math Jam)
• USAMOO 2020 (PDF) (TeX) (video)
• USAMO 2021 (PDF) (TeX) (video)
• USAMO 2022 (PDF) (TeX)
• USAMO 2023 (PDF) (TeX)
• JMO 2010 (PDF) (TeX)
• JMO 2011 (PDF) (TeX)
• JMO 2012 (PDF) (TeX)
• JMO 2013 (PDF) (TeX)
• JMO 2014 (PDF) (TeX)
• JMO 2015 (PDF) (TeX) , featuring Steve !
• JMO 2016 (PDF) (TeX)
• JMO 2017 (PDF) (TeX)
• JMO 2018 (PDF) (TeX)
• JMO 2019 (PDF) (TeX) (Math Jam)
• JMOO 2020 (PDF) (TeX) (video)
• JMO 2021 (PDF) (TeX) (video)
• JMO 2022 (PDF) (TeX)
• JMO 2023 (PDF) (TeX)

USA TST Selection Test (TSTST) #

For an explanation of the name, see the FAQ on the USA IMO team selection .

• TSTST 2011 (probs) (sols) (TeX)
• TSTST 2012 (probs) (sols) (TeX)
• TSTST 2013 (probs) (sols) (TeX)
• TSTST 2014 (probs) (sols) (TeX)
• TSTST 2015 (probs) (sols) (TeX)
• TSTST 2016 (probs) (sols) (TeX)
• TSTST 2017 (probs) (sols) (TeX)
• TSTST 2018 (probs) (sols) (TeX) (stats)
• TSTST 2019 (probs) (sols) (TeX) (stats)
• (video 1) (video 2) (video 3)
• TSTST 2021 (probs) (sols) (TeX) (stats)
• TSTST 2022 (probs) (sols) (TeX) (stats)
• TSTST 2023 (probs) (sols) (TeX) (stats)

USA Team Selection Test (TST) #

These exams are used in the final part of the selection process for the USA IMO team.

• USA Team Selection Test 2000 (probs)
• USA Team Selection Test 2001 (probs)
• USA Team Selection Test 2002 (probs)
• USA Winter TST 2012 (probs)
• USA Winter TST 2013 (probs)
• USA Winter TST 2014 (probs) (sols) (TeX)
• USA Winter TST 2015 (probs) (sols) (TeX)
• USA Winter TST 2016 (probs) (sols) (TeX)
• USA Winter TST 2017 (probs) (sols) (TeX)
• USA Winter TST 2018 (probs) (sols) (TeX) (stats)
• USA Winter TST 2019 (probs) (sols) (TeX) (stats)
• USA Winter TST 2020 (probs) (sols) (TeX) (stats)
• USA Winter TST 2021 (probs) (sols) (TeX) (stats) (video)
• Because of the pandemic, there was no USA Winter TST for IMO 2022.
• USA Winter TST 2023 (probs) (sols) (TeX) (stats)

International Math Olympiad #

• IMO 1997 (PDF) (TeX)
• IMO 1998 (PDF) (TeX)
• IMO 1999 (PDF) (TeX)
• IMO 2000 (PDF) (TeX)
• IMO 2001 (PDF) (TeX)
• IMO 2002 (PDF) (TeX)
• IMO 2003 (PDF) (TeX)
• IMO 2004 (PDF) (TeX)
• IMO 2005 (PDF) (TeX)
• IMO 2006 (PDF) (TeX)
• IMO 2007 (PDF) (TeX)
• IMO 2008 (PDF) (TeX)
• IMO 2009 (PDF) (TeX)
• IMO 2010 (PDF) (TeX)
• IMO 2011 (PDF) (TeX)
• IMO 2012 (PDF) (TeX)
• IMO 2013 (PDF) (TeX)
• IMO 2014 (PDF) (TeX)
• IMO 2015 (PDF) (TeX)
• IMO 2016 (PDF) (TeX)
• IMO 2017 (PDF) (TeX)
• IMO 2018 (PDF) (TeX)
• IMO 2019 (PDF) (TeX)
• IMO 2020 (PDF) (TeX) (video)
• IMO 2021 (PDF) (TeX)
• IMO 2022 (PDF) (TeX)
• IMO 2023 (PDF) (TeX)

Also listed on the USEMO page .

• (video 1) (video 2)
• USEMO 2022 (problems) (solutions+results)

See also general ELMO information .

• ELMO 2010 (problems) (solutions)
• ELMO 2011 (problems) (solutions)
• ELMO 2012 (problems)
• ELMO 2013 (problems) (solutions) (shortlist) (中文)
• ELMO 2014 (problems) (solutions) (shortlist)
• ELMO 2016 (problems) (solutions) (ELSMO)
• ELMO 2017 (problems) (shortlist) (ELSMO) (ELSSMO)
• ELMO 2018 (problems) (shortlist) (ELSMO)
• ELMO 2019 (problems) (shortlist) (ELSMO)
• ELMO 2020 (problems) (ELSMO)
• ELMO 2021 (problems) (ELSMO)
• ELMO 2022 (problems) (ELSMO)

Taiwan Team Selection Test #

These are the problems I worked on in high school when competing for a spot on the Taiwanese IMO team. These problems are in Chinese; English versions here .

• Taiwan TST 2014 Round 1 (problems)
• Taiwan TST 2014 Round 2 (problems)
• Taiwan TST 2014 Round 3 (problems)

NIMO / OMO #

In high school, I and some others ran two online contests called NIMO (National Internet Math Olympiad) and OMO (Online Math Open). Neither contest is active at the time of writing (April 2021) but I collected all the materials and put them in a Google Drive link since the websites for those contests is not currently online. Most of the problems are short-answer problems.

The acronym stands from “math olympiad hardness scale”, pun fully intended .  ↩

Math Olympiad training handouts

• Handouts from Canadian IMO Training camps

Book recommendations

I have taught classes at various math olympiad training programs. Here are some of my handouts and training material.

If you don’t know where to start, I recommend Cyclic Quadrilaterals—The Big Picture and Three Lemmas in Geometry .

• Integer Polynomials - MOP 2007 Black group Integer polynomials, including various irreducibility criteria.
• Inequalities - Canadian 2008 Winter Training Contains a short essay discussing the IMO 2001 inequality.
• Polynomials - Canadian 2008 Summer Training Advanced techniques in polynomials. Roots of unity, integer divisibility, intermediate value theorem, Lagrange interpolation, Chebyshev polynomials, irreducibility criteria, and Rouché’s theorem.
• Determinants: Evaluation and Manipulation - MIT UMA Putnam Talk
• Linear algebra tricks for the Putnam - MIT UMA Putnam Talk

Combinatorics

• Algebraic Techniques in Combinatorics - MOP 2007 Black Group Applications of linear algebra and posets to olympiad-style combinatorics problems.
• Tiling - MOP 2007 Blue group Discussion of tiling boxes with bricks. Contains many coloring and tiling problems.
• Counting in Two Ways - MOP 2007 Blue and Black group
• Combinatorics: bijections, catalan numbers, counting in two ways - Canadian 2008 Winter Training
• Combinatorics: pigeonhole principle, coloring, binomial coefficients, bijections - AwesomeMath 2007
• Combinatorics: counting in two ways, generating functions, algebraic combinatorics - AwesomeMath 2007
• Lemmas in Euclidean Geometry - Canadian 2007 Summer Training A collection of commonly occuring configurations in geometry problems.
• Cyclic Quadrilaterals – The Big Picture - Canadian 2009 Winter Training Explores many properties of the complete cyclic quadrilateral and its Miquel point, and also discusses several useful geometric techniques.
• Three Lemmas in Geometry ( Solutions ) - Canadian 2010 Winter Training
• Power of a Point ( Solutions ) - UK Trinity Training 2011 (Mint group)
• Circles - Canadian 2008 Summer Training Contains a section on a particular tangent circle configuration, and another section on projective geometry, poles and polars. Here ’s some additional food for thought.
• Similarity - Canadian 2007 Summer Training Applications of similar triangles and spiral similarity.

Number theory

• a n ± 1 ( Solutions ) - UK Trinity Training 2011 Working with expression of the form a n ± 1 and the exponent lifting lemma.
• Modular arithmetic: Divisibility, Fermat, Euler, Wilson, residue classes, order - AwesomeMath 2007

Here are some of my book recommendations for preparing for math competitions, in roughly increasing levels of difficulty.

Introductory

• Lehoczky and Rusczyk, The Art of Problem Solving, Volume 1: the Basics
• Lehoczky and Rusczyk, The Art of Problem Solving, Volume 2: and Beyond
• Zeitz, The Art and Craft of Problem Solving
• Engel, Problem Solving Strategies
• Andreescu and Enescu, Mathematical Olympiad Treasures
• Andreescu and Gelca, Mathematical Olympiad Challenges
• Andreescu and Dospinescu, Problems from the Book
• Andreescu and Dospinescu, Straight from the Book
• Djukić et al., The IMO Compendium (complete collection of IMO shortlist problems)

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IMO-AG-30 refers to a set of 30 classical geometry problems adapted from the International Mathematical Olympiad (IMO) contests. These problems are specifically designed to challenge and test the problem-solving abilities of high-school mathematicians in the field of geometry.

Recently, AlphaGeometry , an AI system developed by Google DeepMind, demonstrated exceptional prowess in solving these geometry problems. In a benchmark test, AlphaGeometry solved 25 out of the 30 IMO-AG-30 problems within the standard Olympiad time limits. This achievement places it on par with the average score of human gold medalists who participate in the IMO geometry competitions¹²³.

AlphaGeometry adopts a neuro-symbolic approach , combining a neural language model with a symbolic deduction engine. This powerful combination allows it to find proofs for complex geometry theorems, bridging the gap between AI reasoning and human-level performance in mathematics¹.

The development of AlphaGeometry represents a significant milestone in advancing deep mathematical reasoning and has implications for the future of AI systems across various domains. The code and model for AlphaGeometry have been open-sourced, encouraging further exploration and innovation in mathematics, science, and AI¹⁵.

As Ngô Bảo Châu, a Fields Medalist and IMO gold medalist, aptly put it: "It makes perfect sense to me now that researchers in AI are trying their hands on the IMO geometry problems first because finding solutions for them works a little bit like chess in the sense that we have a rather small number of sensible moves at every step. But I still find it stunning that they could make it work. It's an impressive achievement."¹.

(1) AlphaGeometry: An Olympiad-level AI system for geometry. https://deepmind.google/discover/blog/alphageometry-an-olympiad-level-ai-system-for-geometry/. (2) Inside AlphaGeometry: Google DeepMind’s New Model that Solves Geometry .... https://pub.towardsai.net/inside-alphageometry-google-deepminds-new-model-that-solves-geometry-problems-like-a-math-dad37976fc39. (3) AlphaGeometry: Discover AI System That Solves Geometry Olympiad .... https://hyscaler.com/insights/alphageometry-ai-system-solves-geometry/. (4) GitHub - google-deepmind/alphageometry. https://github.com/google-deepmind/alphageometry. (5) Inside AlphaGeometry: Google DeepMind’s New Model that Solves.... https://towardsai.net/p/machine-learning/inside-alphageometry-google-deepminds-new-model-that-solves-geometry-problems-like-a-math-olympiad-gold-medalist.

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More From Forbes

The keystone of success: trust over transactions in business.

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Pascal Bachmann, founder and CEO of Strategy Achievers .

In the dynamic arena of entrepreneurship, the relentless pursuit of sales and leads often overshadows a fundamental element essential for enduring success: trust. This oversight not only diminishes the value of customer relationships but also hampers long-term growth. In an age where authenticity is increasingly becoming the currency of value, businesses that pivot from a transaction-focused approach to one that prioritizes trust stand out as beacons of success.

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The Problem With Sales-Centric Strategies

The traditional sales model encourages entrepreneurs to focus on closing deals, often at the expense of understanding and solving actual client needs. This model fosters superficial interactions, where discovery calls are masked sales pitches rather than genuine attempts to understand a potential client's challenges and aspirations. Such interactions lack depth and fail to create meaningful connections, rendering them ineffective in building long-lasting business relationships.

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At its core, building trust necessitates a sincere interest in understanding and solving clients' problems. This begins with reframing discovery calls as opportunities to delve deep into clients' worlds without any ulterior motive to sell. By adopting this mindset, conversations become more authentic and less awkward, facilitating a natural exchange of energy that either feels right or doesn't.

Strategy achievers leverage this principle by implementing a personalized framework tailored to uncover these insights while staying true to their core values. Their process involves getting to know clients on a personal level first—understanding how they wish to be perceived by their target audience, identifying who exactly they serve and elucidating the specific problem that can be addressed through their solution strategy.

This authentic engagement demonstrates authority and expertise without overtly selling anything. It establishes trust, which is pivotal for clients when deciding whether or not to invest in a service or product. When there's clarity about the gap between their current state (problem) and desired state (solution) and how your offer perfectly bridges that gap (solution), price becomes secondary to results.

Solution: A Shift In Perspective

The essence lies in focusing on solving problems rather than merely closing clients. Creating an authentic framework for engagement allows businesses to live out their core values transparently while ensuring every team member is aligned with this philosophy.

Such frameworks aren't just methodologies; they're reflections of genuine interest in client success beyond financial transactions. They embody an ethos where selling isn't forced but becomes a natural progression because clients genuinely want what’s being offered due to established trust.

Implementing Authentic Engagement Frameworks

Cultivating trust at every interaction point with potential clients requires systematic yet authentic frameworks. Personally, I follow this framework, and I recommend you give it a try:

1. Personal Connection: Always start by understanding the person behind the business or need.

2. Perception Analysis: Dive into how they want to be perceived by their target audience.

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2015 IMO problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. (In Thailand)

• Problem 1 proposed by Merlijn Staps, Netherlands
• Problem 2 proposed by Dušan Djukić, Serbia
• Problem 3 proposed by Danylo Khilko and Mykhailo Plotnikov, Ukraine
• Problem 4 proposed by Silouanos Brazitikos and Evangelos Psychas, Greece
• Problem 5 proposed by Dorlir Ahmeti, Albania
• Problem 6 proposed by Ross Atkins and Ivan Guo, Australia
• IMO Problems and Solutions, with authors
• Mathematics competition resources

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