These are FAQ's about the rules of math contests (e.g. MOP qualification, IMO criteria, and so on) that I often get asked. Please note: this is not an official source in any way; it represents my best knowledge and may become out of date. See the official AMC website for authoritative answers.
If you are looking for training advice, see Contest FAQ's.
1. How is the USA IMO team determined?
Starting in 2011 and as of 2019, the IMO selection procedure is as follows.
In the USA the main olympiad event is the USA(J)MO held in April each year. Roughly 250 students qualify for the USAMO and 250 students qualify for the USA(J)MO via scores on previous short-answer contests. Then things get interesting:
- In USAMO of year $N-1$, about 60 students in grades 11 and below, as well as the IMO team of that year, are invited to the three-week Mathematical Olympiad Summer Program, abbreviated MOP.
- At the end of MOP, an IMO-style exam called the TSTST (yes, I know) is given. The acronym stands for "Team Selection Test Selection Test". The top approximately 30 students are then selected for the year-round TST's for the IMO of year $N$.
- Two TST's are given in December and January, administered by schools. Each of the two exams is worth 21 points.
- Contestants participate in RMM Day 1, worth 21 points.
- Contestants participate in the APMO, which is weighted by 0.6; hence it is also worth 21 points (despite being a five-problem exam).
- Finally, the two days of USAMO of year $N$ serves as the final TST, worth 42 points.
- The sum of scores over these five tests (six days, maximum $126$ points) determine that year's IMO team.
Hence the IMO selection process takes an entire year. The IMO team is known shortly after the USAMO grading in May.
Note in particular that winning the USAMO is neither necessary or sufficient for qualifying for the American IMO team, since the IMO team is determined by several different exams.
2. How is the Taiwan IMO team determined?
Here are the details for IMO 2014 (the year which I participated), A series of tests/camps take place and at each step the number of contestants is reduced. Interestingly, the Taiwan national olympiad is not involved in the selection process at all.
- The first test worth mentioning is the APMOC, an exam in the same format as the APMO.
- High scores on the APMOC are invited to another camp at which the APMO is given in March.
- The top 30 scores on the APMO are invited to the first TST camp in late March.
- The top 15 scores at the first TST camp are invited to the second TST camp two weeks later.
- The top 10 scores at the second TST camp are invited to the third TST camp two weeks later.
- The IMO team is decided by a weighted sum of the scores at the second and third TST camp plus an oral interview.
Each camp lasts roughly from Friday to Tuesday; the geography of Taiwan makes commuting very convenient. Each TST camp features
- Three Team Selection Quizzes (formally "individual study"), which last 110 minutes with two olympiad problems
- A full mock IMO with six problems.
Thus the IMO team is decided by $6 \cdot 2 = 12$ mock IMO problems and $2 \cdot 3 \cdot 2 = 12$ quiz problems. The weighting is roughly so that a mock IMO problem is worth three quiz problems. The selection process takes a few months. The team is known shortly after the third camp, a couple days into May.
For more details, see my report on Taiwan IMO.
3. How is USAMO / IMO graded?
Don't take my word for it: try reading the USAMO 2003 rubric.
In general, for each problem the solution is graded according to the rubric:
- 7: Problem was solved.
- 6: Tiny slip (and contestant could repair)
- 5: Small gap or mistake, but non-central
- 2: Lots of genuine progress
- 1: Significant non-trivial progress
- 0: "Busy work", special cases, lots of writing
- There is a clear distinction between success/failure, and the scores 3, 4 are absent (they are awarded rarely, most usually in two-part problems).
- There is no style: a correct solution receives 7 points. However, it is still in your best interest to write a solution clearly to minimize the chance the grader doesn't understand you.
- It is hard to earn partial credits with incomplete solutions and easy to earn full credit for complete solutions. In particular, 0 is the most common score for non-complete solutions.
The grading procedure itself is also different for USAMO / IMO. At the USAMO, we grade each paper twice before assigning a score, then re-grade approximately the top 25% of papers again. At the IMO, a process called coordination is used in which team leaders defend their contestant's solutions to coordinators.
4. Can I get a regrade on USA(J)MO?
It is a long-standing policy of AMC that decisions of the USAMO graders are final and may not be appealed. I am also not able to look up your exact score distribution for you.
I won't claim that we graders never make mistakes, but all papers are read independently by at least two graders (and sometimes more if the solution is especially different or unusual); these graders must agree before a score is assigned. Then the top papers on each contest are taken and re-graded again. Nearly all the graders are either past USAMO winners or IMO medalists, so there is no question about grader qualification.
All that being said, this is a good incentive to learn to write your solutions neatly. You don't want to be the one with a correct solution that the graders can't understand!
5. What results may be cited without proof on USAMO or IMO?
There are no hard criteria set in stone, but in general, any well-known result which does not trivialize the problem is okay. If you are still unsure, it is strictly better to include the proof. See English handout for more discussion.
6. How much detail should I include in a proof to avoid losing points?
Enough to convince the grader that you understand the solution to the problem.
Whenever you say something like "it's easy to see X", the grader has to ask themself whether the student actually understands why X is true, or doesn't know and is just "bluffing". In particular, this is a math issue, not a style issue.
As a very loose approximation, the official solutions file for USAMO is about as terse as you can be. See English handout for more discussion.
7. What are the criteria for invitation to MOP?
Before I say anything I want to say that the criteria for MOP invitations are not especially well-defined. Each year, the exact number and choice of students is determined based on the exact scores for that year.
That being said, as of 2016 the criteria for MOP is roughly as follows:
- IMO team members and alternates ("black" group)
- The next approximately 12 non-graduating USAMO students ("blue" group).
- The next approximately 12 USAMO students in 9th and 10th grades ("green" or "red")
- The top approximately 12 students on USAJMO ("red" group)
- Some varying number of non-graduating female contestants from either USAMO or USAJMO (these students represent USA at the European Girls' Math Olympiad). The exact cutoffs for each contest are determined based on the scores for that year.
Young students in 8th grades and below are invited to MOP if and only if the moon is full and the wind is blowing south-south-east. All selection is done by ID number, without student names.
The "color groups" are a convenient shorthand and not worth worrying about.
8. When are ties on USA(J)MO broken?
As of 2016, ties are generally only broken for USAMO winners, since exactly 12 winners are invited to the state dinner.