Why Americans Should Study the Moscow Math Olympiads

MMO 1993-1999I have already written about how American math competition are illogically structured, for the early rounds do not prepare students for the later rounds. The first time mathletes encounter proofs is in the third level, USAMO. How can they prepare for problems with proofs? My suggestion is to look East. All rounds of Russian math Olympiads — from the local to the regional to the national — are structured in the same way: they have a few problems that require proofs. This is similar to the USAMO. At the national All-Russian Olympiad, the difficulty level is the same as USAMO, while the regionals are easier. That makes the problems from the regionals an excellent way to practice for the USAMO. The best regional Olympiad in Russia is the Moscow Olympiad. Here is the problem from the 1995 Moscow Olympiad:

We start with four identical right triangles. In one move we can cut one of the triangles along the altitude perpendicular to the hypotenuse into two triangles. Prove that, after any number of moves, there are two identical triangles among the whole lot.

This style of problems is very different from those you find in the AMC and the AIME. The answer is not a number; rather, the problem requires proofs and inventiveness, and guessing cannot help.

Here is another problem from the 2002 Olympiad. In this particular case, the problem cannot be adapted for multiple choice:

The tangents of a triangle’s angles are positive integers. What are possible values for these tangents?

MMO 1993-1999

The problems are taken from two books: Moscow Mathematical Olympiads, 1993-1999, and Moscow Mathematical Olympiads, 2000-2005. I love these books and the problems they present from past Moscow Olympiads. The solutions are nicely written and the books often contain alternative solutions, extended discussion, and interesting remarks. In addition, some problems are indexed by topics, which is very useful for teachers like me. But the best thing about these books are the problems themselves. Look at the following gem from 2004, which can be used as a magic trick or an idea for a research paper:

A deck of 36 playing cards (four suits of nine cards each) lies in front of a psychic with their faces down. The psychic names the suit of the upper card; after that the card is turned over and shown to him. Then the psychic names the suit of the next card, and so on. The psychic’s goal is to guess the suit correctly as many times as possible.
The backs of the cards are asymmetric, so each card can be placed in the deck in two ways, and the psychic can see which way the top card is oriented. The psychic’s assistant knows the order of the cards in the deck; he is not allowed to change the order, but he may orient any card in either of the two ways.
Is it possible for the psychic to make arrangements with his assistant in advance, before the latter learns the order of the cards, so as to ensure that the suits of at least (a) 19 cards, (b) 23 cards will be guessed correctly?
If you devise a guessing strategy for another number of cards greater than 19, explain that too.



  1. Joshua Zucker:

    Hi Tanya,
    We’re moving in this direction in the US, too. There’s been the Colorado olympiad for more than a decade, and now the Bay Area Math Olympiad (https://bamo.org) is about to have its 14th exam. For the last several years, BAMO has also had a separate exam for kids up to grade 8, so they can get an early start on these kinds of problems. I’m sure there are other regional exams like these but I don’t know about them.

    ARML has long had the power round and a power contest once or twice a year that kids can do at their schools. The Mandelbrot contest also has an excellent team round involving proof-writing.

    There are also tests like the USAMTS that help kids prepare for this kind of thinking and writing.

  2. Tatiana Shubin:

    Hi Tanya,

    There’s one more competition MAA (and AMC, in particular) is trying to spread around the US – Math Wrangle, an adaptation of the Russian Math Battle. It requires not only solving problems that require proof but also presenting solutions and criticizing solutions presented by others. This competition can be equally well adopted for high school and middle school level and for more or less advanced kids, too. Math Wrangles were demonstrated at a number of national meetings – JMMs 2010, 2011, and 2012, and MathFest 2011, and a number of regional meetings, and they will be staged again at the MathFest in Madison this year and at JMM 2013 in San Diego, plus at the annual conference of the National Association of Math Circles – Circle on the Road – in Washington, DC, April 13-15, 2012. Math Wrangle rules and problem sets can be found at https://sigmaa.maa.org/mcst/

  3. Bob:

    I’ve participated in Academic Games, and part of it are two games, Equations and OnSets. One is about solving equations, the other is about set theory. These are not like traditional mathematics competitions, in these games you set up a problem and try to come up with a solution before your opponent. A complete game is typically finished in less than 30 minutes, and each game has several rounds, but the games force you to think on your feet and are more engaging than other math competitions in my opinion.

    I have the most experience with Equations, and you come across diverse topics such as prime factorization, modular arithmetic, and change of base. You set up a few restrictions for a round, arrange the number/operations cubes to form an expression, then you and your opponents go back and forth putting cubes in play until someone can come up with a solution using the cubes. You can also claim that no solution is possible. When it comes time to write a solution, you must be able to prove it to your opponents, or disprove an opponent’s solution. You’re not required to write a full “rigorous” proof, just enough to show you’re correct, but in the training for these games students are encouraged to learn theorems and prove them to come up with strategies they can use. These games definitely helped me think about math differently and I personally think they have the best chance of getting many kids interested in math, instead of just the kids in advanced math classes. The only downside is that these games are not that widespread. National competitions typically have teams from Florida, Louisiana, and Michigan and don’t have very many teams from other states.

  4. Asher:

    The 1995 problem turns out to be a very fun and beautiful proof. I enjoyed solving it.

    Also, the American Regions Math League (which mathletes from dozens of states participate in) features a “Power Question”, which is a multi-hour problem involving several proofs and questions on a single topic. It is very exciting.

  5. Konstantin:

    Таня, поскольку все равно часть читающих Ваш блог владеет русским, я напишу по-русски, – иначе мне будет тяжело выразить все мысли, которые я хочу сформулировать.

    1) Поздравляю Вас с 8 Марта!
    2) Мне кажется невозможным ввести в олимпиадную практику задачи на доказательство, если таких задач практически нет при обучении в школе. Это как пропасть в два прыжка преодолевать.
    3) По моему убеждению, в России есть три олимпиады с превосходными задачами, и я затрудняюсь сказать, какая из этих олимпиад лучше. Во-первых, это Турнир Городов (https://turgor.mccme.ru/). Во-вторых, Санкт-Петербургская городская олммпиада (https://www.pdmi.ras.ru/~olymp/). И в-третьих, Московская. (https://olympiads.mccme.ru/mmo/). Кстати, московская олимпиада вот уже три года не является этапом Всероссийской. Так получилось. У нас тут свои олимпиадные тараканы…

  6. Konstantin:

    About 36 cards trick:
    https://ru-math.livejournal.com/114368.html (Russian text)

  7. Tanya Khovanova’s Math Blog » Blog Archive » Guessing the Suit:

    […] recently published my new favorite math problem: A deck of 36 playing cards (four suits of nine cards each) lies in front of a psychic with their […]

  8. Vera:

    I fully agree with the statement that AMC and AIME on one hand, and USAMO on the other, require completely different skills. I have a personal stake in it – my own kids were taught partly Russian style, and are good with mathematical proofs, but tend to make stupid mistakes in 4-minutes AMC problems. My daughter finally managed to get to the USAMO only in the 12th grade, and made the Honorable Mention (result in the second dozen). Now every winter my son wastes his time practicing these 4-minute problems, instead of working on a difficult one (with the same unimpressive results. But what upset me more was that in case he got to the USAMO he would not do well there either, as he spent all his time on easy tasks). This year we are trying an alternative way: through USA MTS. They have nice problems, and a high score allows to get to the AIME bypassing AMC. However I do not understand how many points one gets – if the maximum is 75, it is useless.
    At the same time I understand the attractiveness of an Olympiad with an easy first round – it allows not-so-well-trained children to try and enjoy. This is the reasoning behind competitions like “Kangaroo”, and it makes sense. Not to mention that they simply do not teach mathematical proof at the US schools, and that I cannot imagine a first-round American test other than multi-choice. I believe that the US schema worked well enough for some time, while the set of children quickly solving easy problems was close to that who could do mathematical proofs. I am not sure this is still the case, partly because there now is a huge group of children trained to do an AMC problem in 3 minutes. They still do not do well on complex problems, but it is extremely difficult to compete with them in the first and second rounds.

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