AIME and USA(J)MO instructions
After the AMC 10 and AMC 12 come the invitational and olympiad stages of the pathway. The AIME identifies the strongest scorers; the USAMO and USAJMO turn the very best of them toward the United States team for the International Mathematical Olympiad.
Students who perform well on the AMC 10 or AMC 12 are invited to continue in the AMC series of examinations, a sequence that ultimately leads to the International Mathematical Olympiad. The route is the same for everyone, and each rung is harder than the last:
AMC 10 / AMC 12 → AIME → USA(J)MO → IMO.
This page explains the two stages that sit above the AMC 10/12: the American Invitational Mathematics Examination, and the two olympiads that follow it.
The AIME
The American Invitational Mathematics Examination (AIME) is the invitational stage reached by the highest scorers on the AMC 10 and AMC 12. It is a single 15-question paper, three hours long, and every answer is an integer between 0 and 999, so no answer choices are given and no calculator is used. The questions are markedly harder than those on the AMC 10 and AMC 12, and they are designed to separate the strongest students from one another.
Qualification depends on how each AMC 10/12 paper turns out, but as a guide at least the top 2.5% of AMC 10 scorers and at least the top 5% of AMC 12 scorers are invited. The AIME is offered on two dates — AIME I and AIME II — so that a student who cannot sit the primary date, or who qualifies through a later AMC paper, still has a sitting available. The highest-scoring participants on the AIME are then invited to the USAMO or USAJMO.
The USAMO and USAJMO
The United States of America Mathematical Olympiad (USAMO) and the United States of America Junior Mathematical Olympiad (USAJMO) are the proof-based olympiads that follow the AIME. Each is a six-problem examination written over two days, nine hours in total, in which students must construct full written proofs rather than supply numerical answers.
Students reach these olympiads through a combined index of their AMC and AIME scores, not the AIME alone. The highest-scoring AMC 12 participants, ranked by an AMC 12 and AIME combination, are invited to the USAMO; the highest-scoring AMC 10 participants, ranked by an AMC 10 and AIME combination, are invited to the USAJMO. The two indices are:
- USAMO Index = AMC 12 Score + 20 × AIME Score
- USAJMO Index = AMC 10 Score + 20 × AIME Score
The USA(J)MO is where the national team begins to take shape. Top performers are invited to the Mathematical Olympiad Program in the summer that follows, and participants from that program are then eligible to be selected for the six-person team that represents the United States at the IMO the following summer.
AIME and USA(J)MO at a glance
| AIME | USA(J)MO | |
|---|---|---|
| Format | 15 questions; each answer an integer from 0 to 999; no calculator | 6 problems; full written proofs |
| Length | 3 hours, one sitting (AIME I or AIME II) | 9 hours across two days |
| Who qualifies | Top AMC 10/12 scorers (about the top 2.5% on AMC 10, top 5% on AMC 12) | Top scorers by combined AMC + AIME index (USAMO from AMC 12, USAJMO from AMC 10) |
In short, the AIME is the gateway and the USA(J)MO is the proof-based summit: clear the AMC 10/12, qualify for the AIME, and a strong combined index opens the olympiad and, for a handful of students each year, the path to the IMO.