Evan Chen《陳誼廷》

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Math olympiad beginner's page

This page is meant for people who don’t have much past olympiad/proof experience and are looking to get started. If you aren’t interested in proof-based problems yet, then this page is not for you. Try checking FAQ C-0 if you are totally new to math contests.

Before all else, welcome to the olympiad scene! It’s going to be hard as heck, but in my private opinion this is where all the coolest stuff is (as far as math contests go, anyway 1 ). Stay around for long enough, and you will get to see a lot of really amazing problems.

You may also read math contest FAQs for some more philosophical (and less concrete) advice on studying for math contests in general.

0. Syllabus #

I wrote an unofficial syllabus for math olympiads (also linked on handouts ) giving some guidance on what topics appear on math olympiads.

1. First reading: the welcome letter #

For the USA Math Olympiad in 2020, the board of the USAMO prepared an invitation letter 2 for all the qualifiers, congratulating them on their achievement and giving them some suggestions on where to begin. This letter contains:

  • A few pretty carefully chosen problems (not necessarily easy!), to give people a sense of what to expect on the contest
  • Fully written solutions to those chosen problems, so that you can see what a correct and complete solution is expected to look like.
  • Some advice for actually taking the contest: the format of the exam, planning your time, common mistakes, etc.

You can download the letters here:

  • JMO welcome letter , and solutions to examples
  • USAMO welcome letter , and solutions to examples

I suggest starting by reading through this letter, trying the example problems (you will probably not solve them all; we chose examples from the entire difficulty spectrum), and then comparing your work to the provided solutions. That will give you a taste of what you are getting in to.

2. Writing proofs #

If you don’t have experience with proof-based problems, the first thing I should say is that it is not as hard as you might think . Solving the problems completely is difficult, but if you really have a completely correct solution to a problem, it is actually pretty hard to not get full credit. I would say at least 90% of the time, when a student loses points on the USA(J)MO unexpectedly, it’s because their proof is actually incomplete, not (just) badly-written.

Of course, you should still try to write your solutions as clearly as possible. To that end, here are some links to advice:

  • How to write a solution , from Art of Problem Solving.
  • Remarks on English , written by me.
  • How to write proofs , by Larry W. Cusick.

You don’t need to get too caught up in these links; proof-writing will become more natural over time as you solve more problems. So I would encourage you to continue doing practice problems or reading books at the same time as you are getting used to writing; these go hand-in-hand and I actually suspect it’s counterproductive to try to practice writing in isolation.

If possible, the best way is to have a friend or coach who can check your work and provide suggestions. But the supply of people willing to do this is admittedly very low, so most people are not so lucky to have access to feedback. Almost everyone gets by instead with something like the following algorithm:

  • Write up your solution neatly.
  • Look up the problem on AoPS contest index 4 ; and compare your solution to those by reputable users.
  • Edit your solution and post it on the thread. By Cunningham’s Law , wrong solutions are often exposed quite rapidly.

Together these three steps should catch “most” substantial errors. See Section B.1 of my English handout for more details about this procedure.

If you want a book to follow, the one I grew up with was Joseph J Rotman’s Journey into Mathematics: An Introduction to Proofs . 3

If you like excessive information, you might also read my handout Intro to Proofs for the Morbidly Curious .

3. (For USA) United States Mathematical Talent Search and USEMO #

If you are in the United States, there is a nice proof-based contest called the USAMTS which is a great way for beginners to get started. Things that make the USAMTS special:

  • It is a free, individual, online contest open to any students in the USA.
  • The problems are chosen to be quite beginner-friendly, though with a spectrum of difficulty each round.
  • This contest gives you a full month to work on the problems rather than having a short time limit.
  • You get some feedback on your proofs as well, not just a score.

I also run a contest called the USEMO in the fall which is also free and offers feedback, but it is more difficult since is intended to mimic the USAMO and IMO in format and difficulty. One could try using it as “practice USAMO” in the fall.

4. Books to read #

There is some more material you have to learn as well, since there are some new classes of problems (such as olympiad geometry, functional equations, orders mod p, etc.) that you will likely not have seen before from just working on short-answer contests.

Some possible suggestions for introductory books:

  • General: Art and Craft of Problem-Solving by Paul Zeitz is a good “first book” for all the fields.
  • Geometry: My book E.G.M.O. ; and more alternatives are linked at the bottom of that page.
  • Number theory: Modern Olympiad Number Theory is the most comprehensive text I know of now.

The OTIS Excerpts has beginner introductions for several topics:

  • Inequalities (chapters 1-2)
  • Functional equations (chapters 3-4)
  • Combinatorics (chapters 6-9)

More possibilities (including intermediate-advanced texts not listed here) are on the links page . You might also check Geoff Smith’s advice and links .

I really want to stress these are mere suggestions . Just because you have done X does not mean you will achieve your goals, and conversely, there are surely many fantastic resources that I have not even heard of. If you are looking for a list of materials which are “guaranteed to be enough” for solving IMO #1 and #4, then unfortunately I can’t help you.

5. Problem sources #

At some point (sooner rather than later), you also need to start just working through some past problems from recent years of contests. You can think of this as encountering problems in the wild. 5

In case you didn’t know already, on Art of Problem-Solving there is an extensive archive of past problems from basically every competition under the sun, together with community-contributed solutions. The supply of problems here is inexhaustible.

Here are some particular contests I like (alphabetical):

  • Canada national olympiad
  • European Girls Math Olympiad
  • IMO and IMO Shortlist
  • NICE , open to anyone
  • USA Team Selection Tests

The bottom of the recommendations page has some more suggestions for problems if this list isn’t sufficient.

Good luck and happy solving!

Insomuch as contest problems have intrinsic artistic value, proof-based exams are a more versatile medium than the short-answer exams, much like how videos are more versatile than static photos. In this analogy, videos don’t make stills worthless or obsolete; they aren’t automatically better, either. But as a medium, they expand the space of ideas an artist could express, at the cost of being proportionately more work to create.  ↩

These were written in early January 2020 before COVID-19 wreaked havoc on everything, so the contests still go by their typical name and don’t mention anything specific to the belated administration that year.  ↩

It shares my philosophy that teaching proof-based classes by force-feeding set theory notation is not particularly helpful, and instead develops proof-writing by discussing real mathematical content from geometry, number theory, etc. rather than being overly focused on bookkeeping and formalism.  ↩

I do NOT recommend using the AoPS Wiki in place of the Contest Index. The solution quality in the wiki is generally much poorer than the forum.  ↩

I used to carry a binder with printouts of the IMO shortlist and check them off as I solved them.  ↩

art problem solving olympiad resources

About the Art of Problem Solving Initiative

Founded in 2004, the Art of Problem Solving Initiative, Inc. was created by people who love math and love teaching to help students access the study of advanced mathematics.

The Initiative began by running the USA Mathematical Talent Search , a nationwide math contest sponsored by the National Security Agency which continues to run to this day. In each round, this contest gives students a full month to work on five challenging proof-based problems. Students then get individual feedback on their work, including comments on their mathematical reasoning and on their proof-writing skills. By taking away the usual time pressures seen in many math contests, students have the opportunity to go more deeply and to explore.

For many years, the Initiative also ran the Local Programs Initiative which provided fiscal sponsorship to math circles and clubs across the country.

Now, the main project of the Initiative is Bridge to Enter Advanced Mathematics (BEAM) , a project to help underserved students find a realistic pathway towards becoming scientists, mathematicians, engineers, and programmers. During the summer after 7th grade, students are invited to participate in a free three-week residential program on a college campus where they gain the academic and social/emotional preparation to succeed in future programs for advanced study. They then receive academic advising throughout 8th grade and high school to help them and their families find the best opportunities for their educations.

The Art of Problem Solving Initiative receives support from Art of Problem Solving (AoPS) , which develops resources for high-performing middle and high school students including the largest online community of avid math students in the English-speaking world. The AoPS online school has trained many winners of major national mathematics competitions, including several gold medalists at the International Math Olympiad, Davidson Fellows, and winners of the Intel and Siemens Talent Search competitions.

The Art of Problem Solving Initiative is a 501(c)(3) charitable organization. It continues to be led by expert mathematicians and educators to this day. Please consider joining our contributors by donating to support advanced math education.

Financial Information

The Art of Problem Solving Initiative, Inc. makes its financial statements available for public inspection in the interest of greater transparency. Please see below to learn more about the organization.

  • Tax return (Form 990)
  • Audited financial statements
  • Tax return (Form 990) (Note that due to a change in fiscal year these cover a shortened period.)

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Art of problem solving brings its problem-solving teaching methods to local academic centers, with one virtual campus and 15 locations in the us, and more being added every year., we challenge students.

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Art of Problem Solving has been a leader in math education for high-performing students since 1993. We launched AoPS Academy in 2016 to bring our rigorous curriculum and expert instructors into classrooms around the United States. With campuses in 8 states (and growing!), our approach nurtures a love for complex problem solving, which is fully incorporated into all our math, science, and language arts courses. Most importantly, our students become part of a community of motivated learners that helps elevate them to new heights.

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Our family regularly talks about AoPS. We try to think about how our life would be different without you all. What if my daughter hadn't learned to love math? What if she never experienced being pushed to her limits? Overcoming failure? She wouldn't be who she is. And she is AWESOME! A very proud mom here. We are so fortunate that we found AoPS Academy. I credit AoPS for much of her confidence.

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Art of Problem Solving textbooks have been used by outstanding students since 1993. The AoPS website launched in 2003, and its online community now has over 300,000 members. Many of the winners of each year’s International Math Olympiad use the AoPS site as a primary training resource. The AoPS online school has over 15,000 enrollments annually in courses specifically designed for high-performing math students. Most of the winners of major American national math competitions are AoPS alumni, and thousands of our alumni enroll in top universities each year.

With AoPS Academy, Art of Problem Solving brings its curriculum and pedagogical techniques to the classroom.

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Getting the most out of the UK Chemistry Olympiad

  • 2 Your complete guide to the UK Chemistry Olympiad
  • 3 What you and your students will gain from the Olympiad
  • 4 ‘It’s a really enjoyable experience and good preparation for A-level’
  • 5 Join the problem-solving set
  • 6 ‘Having a go at something really difficult builds confidence’
  • 7 Questions, questions, questions …

Join the problem-solving set

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Top tips from two heads of science on how to run a successful Olympiad club

A chemistry teacher

Source: © Olivia Waller/Folio Art

Let pupils find their own groups, but be ready to intervene and prompt them to find solutions

Meet Helen Alonzi and Ryan Beattie, heads of science at secondary schools in the UK who’ve both entered students into the UK Chemistry Olympiad and been rewarded with success. In this article they share their top tips on how to set up and run extracurricular Olympiad clubs and their favourite resources.

Helen’s top tips

‘There’s been an increase in students wanting to do the Olympiad and applying for chemistry at university, so we set up a problem-solving club,’ explains Helen Alonzi, who introduced her school’s club in 2012. ‘The focus is from September to December, but it runs all year in a classroom after school, with about a dozen 16 to 18 year-old learners. Some come throughout; others drop off after Round one.’

1. Spread the word

We promote the club via a poster for classroom doors, a list of all co-curricular clubs on the school website for students and parents, and as one of the general notices that go to all year 12 students (via our homework setting app) at the start of the year.

2. Mix it up

It’s a mixed group. Sometimes the older year peer-teach the younger, sometimes I give the same question to all, knowing the more confident will work at their own speed while I use whiteboards to help the others. This year we had some particularly strong year 13 chemists, who worked through a bank of past Olympiad papers to produce model answers that a teacher could use in future.

3. Manage group work

Pupils naturally fall into their own groups; it’s best to let them get started and intervene if they need prompts. I go through solutions with individual groups once they’ve completed small sections. The more confident like to have access to a mark scheme so they can check themselves. In the early days, less confident students may not get through much of a question, and that’s fine – it promotes resilience to just keep trying each week.

4. Use the explainers

When I first started looking at Olympiad questions, the explainers gave me confidence to talk about less familiar topics. Now, I direct students to them, especially when Round one is approaching. Cambridge Chemistry Challenge papers are good for running a problem-solving class – the questions are written for year 12 so tend to be shorter and accessible.

When I first started looking at Olympiad questions, the explainers gave me confidence to talk about less familiar topics. Now, I direct students to them, especially when Round one is approaching. Cambridge Chemistry Challenge papers ( bit.ly/3UI20OQ ) are good for running a problem-solving class – the questions are written for year 12 so tend to be shorter and accessible.

Helen’s favourite online resources

When students can’t attend the club, Helen directs them to:  How to prepare for the Chemistry Olympiad , the Chemistry Olympiad support booklet , Chemistry Olympiad introductory questions and Chemistry Olympiad explainers .

When students can’t attend the club, Helen directs them to: How to prepare for the Chemistry Olympiad ( rsc.li/4dNDsN6 ), the Chemistry Olympiad support booklet ( rsc.li/3UTcYB4 ), Chemistry Olympiad introductory questions ( rsc.li/4bLNXi3 ) and Chemistry Olympiad explainers ( rsc.li/3yDpayj ).

A boy with a chemistry symbol on his t-shirt with a tennis racket

Finding the right level is important. Chunking, mini whiteboards and reteaching all have a part to play

Ryan’s top tips

‘Our club begins in October and ends in December, with five timetabled, fortnightly hour-long workshops,’ says Ryan Beattie, who oversees about 15 chemistry students a year entering the Olympiad. ‘It includes explicit idea instruction, followed by modelling of answers, content chunking and whiteboards to check understanding.’

1. Get started

Promotion is on an online learning platform, like Moodle or Google Classroom, with an overview of the UK Chemistry Olympiad , date and time of the first session.

I use flipped learning before sessions – a YouTube video, RSC resource or accessible question from Round one papers – plus signposted questions relating to the area we’ve covered at the end.

2. Choose your topic

We choose from topics assessed by Round one that aren’t on the A-level specification. I use Olympiad explainers and for deeper understanding of organic mechanisms, for example, we might look at curly arrows .

We choose from topics assessed by Round one that aren’t on the A-level specification. I use Olympiad explainers and for deeper understanding of organic mechanisms, for example, we might look at curly arrows ( rsc.li/4bQnJv3 ).

3. Find the right level

Pitching the level can be tricky, so I deliver content in small chunks and check understanding after each. Students write responses on mini whiteboards and reveal them at the end of a countdown, allowing me to identify anyone struggling. A quick reteach is often all that’s needed. I set pupils independent practice and get them to collaborate in groups of two or three. High achieving students use the mark scheme to check their answers after they’ve completed a number of questions, leaving me free to support others. At the end, I go through solutions using a visualiser.

Get Ryan’s catalogue of questions

‘I’ve created a user-friendly catalogue of the Olympiad questions from previous Round one papers, loosely split by topic, with links to key resources, says Ryan. ‘It came about for students wanting to work on specific topics after meetings and saves me time in planning sessions. I can use questions in the session and signpost others to practise independently later.’

‘I’ve created a user-friendly catalogue of the Olympiad questions ( bit.ly/3yrvHw1 ) from previous Round one papers, loosely split by topic, with links to key resources,’ says Ryan. ‘It came about for students wanting to work on specific topics after meetings and saves me time in planning sessions. I can use questions in the session and signpost others to practise independently later.’

Lynne Maxwell

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Mid Michigan Olympiad in Mathematics

Mid Michigan Mathematical Olympiad is a proof-based competition, held for 3 hours and usually having 5 problems for grades 5-12. It is held in the MSU campus yearly. Some very orz and xooks participants include the all mighty [b] mahaler [/b] and [b]sanaops9[/b].

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  1. Resources

    Art of Problem Solving offers free resources for avid problem solvers, including games, Alcumus, math videos, the AoPS Wiki, and a LaTeX tutorial. ... Class Schedule Recommendations Olympiad Courses Free Sessions/Math Jams Video Classes books tore ... Art of Problem Solving is an ACS WASC Accredited School. aops programs. AoPS Online. Beast ...

  2. Art of Problem Solving

    Since 2015, all USA International Math Olympiad teammates have been AoPS students. AS SEEN IN. ... Free Student Resources. Art of Problem Solving offers a wide variety of free resources for avid problem solvers, including hundreds of videos and interactive tools like Alcumus, our popular adaptive learning system. ...

  3. Evan Chen • For beginners

    General: Art and Craft of Problem-Solving by Paul Zeitz is a good "first book" for all the fields. Geometry: My book E.G.M.O. ; and more alternatives are linked at the bottom of that page. Number theory: Modern Olympiad Number Theory is the most comprehensive text I know of now.

  4. Art of Problem Solving

    2022 IMO problems and solutions. The first link contains the full set of test problems. The rest contain each individual problem and its solution. (In Norway) Problem 1; Problem 2; Problem 3; Problem 4; Problem 5; Problem 6; See Also. IMO Problems and Solutions, with authors; Mathematics competition resources

  5. Art of Problem Solving

    ITMO/AITMO ( [Asian] International Teenagers Mathematical Olympiad) Directory of Problems w/ Solutions. Archive from 2005 to 2017. Every two years, except 2007. Mathematics. English. Taiwanese. IWYMIC (The International World Youth Mathematics Intercity Competition) All Problems Since 1999 w/ Solutions.

  6. The Art of Problem Solving Initiative : About : General Info

    The AoPS online school has trained many winners of major national mathematics competitions, including several gold medalists at the International Math Olympiad, Davidson Fellows, and winners of the Intel and Siemens Talent Search competitions. The Art of Problem Solving Initiative is a 501(c)(3) charitable organization.

  7. Art of Problem Solving

    Mathematics Olympiads. Mathematics Olympiads are mathematics competitions. In some countries, mathematics Olympiads refer to all math competitions, while in some countries, including the United States, math Olympiads refer to proof-based math competitions. The International Mathematics Olympiad (IMO) is the most famous math Olympiad.

  8. AoPS Academy

    Art of Problem Solving has been a leader in math education for high-performing students since 1993. We launched AoPS Academy in 2016 to bring our rigorous curriculum and expert instructors into classrooms around the United States. With campuses in 8 states (and growing!), our approach nurtures a love for complex problem solving, which is fully ...

  9. Art of Problem Solving

    1. There is one and only one perfect square in the form. where and are prime. Find that perfect square. 2. and are positive integers. If , find . 3.The fraction, where and are side lengths of a triangle, lies in the interval , where and are rational numbers.

  10. Art of Problem Solving

    Art of Problem Solving. Art of Problem Solving textbooks have been used by outstanding students since 1993. The AoPS website launched in 2003, and its online community now has over 300,000 members. Many of the winners of each year's International Math Olympiad use the AoPS site as a primary training resource.

  11. Math Message Boards FAQ & Community Help

    SeboS wrote: Determine all functions f : R R, such that. for all real numbers x and y. Let be the assertion. The only constant solution is . So let us from now look only for nonconstant solutions. Comparaison of with implies. Comparaison of with implies , qtill true when and so is odd. If and for some , then :

  12. Recommended Resources

    With their books, classes and other free online resources, students develop the skills they need to become successful, creative problem solvers. We recommend that students and teachers use the following tools in their FREE resources section: Alcumus, Mathscount Trainer and Free Videos. Art of Problem Solving. Check it Out.

  13. Art of Problem Solving

    Art of Problem Solving is a website that provides rich resources and tools for students who love math and want to challenge themselves. You can find online courses, textbooks, videos, contests, and more on various topics, from prealgebra to calculus. You can also use Alcumus , a free adaptive learning system that gives you personalized problems and feedback.

  14. How to run a successful Chemistry Olympiad club

    1. Spread the word. We promote the club via a poster for classroom doors, a list of all co-curricular clubs on the school website for students and parents, and as one of the general notices that go to all year 12 students (via our homework setting app) at the start of the year. 2. Mix it up.

  15. SOSE-ARISE Lecture Series: Dr Aberin talks about students' processes in

    On 30 April 30 2024, Dr Maria Alva Q Aberin, Associate Professor of the Department of Mathematics, whose expertise is in the field of mathematics education and discourse analysis, shared her knowledge and profound insights at the 7th session of the School of Science and Engineering - Ateneo Research Institute of Science and Engineering (SOSE-ARISE) Lecture Series.

  16. Mid Michigan Olympiad in Mathematics

    Mid Michigan Mathematical Olympiad is a proof-based competition, held for 3 hours and usually having 5 problems for grades 5-12. It is held in the MSU campus yearly. Some very orz and xooks participants include the all mighty [b] mahaler [/b] and [b]sanaops9 [/b]. Art of Problem Solving is an. ACS WASC Accredited School.