The USAMO (United States of America Mathematical Olympiad) score is calculated based on the total points earned from solving six proof-based problems. Separately, qualification for the USAMO is determined by a USA(J)MO Index derived from your AMC and AIME scores.
Understanding USAMO Contest Scoring
The USAMO is a rigorous, 9-hour, 6-problem mathematical competition spread over two days. Each of the six problems is worth 7 points, making the maximum possible score 42 points.
- Problem Value: Each problem on the USAMO is worth 7 points.
- Total Score: Your final score is the sum of the points obtained on each of the six problems.
- Partial Credit: Judges award partial credit for significant progress towards a solution, correct ideas, or demonstrating a strong understanding of the problem, even if the final proof is incomplete or has minor flaws. This means a student doesn't need to perfectly solve every problem to earn points.
- Proof-Based: Unlike the AMC and AIME, USAMO problems require full written proofs, not just numerical answers. Clarity, rigor, and logical completeness are crucial.
Calculating the USA(J)MO Qualification Index
While the USAMO itself has a specific scoring system, students must first qualify to participate. Qualification for the USAMO (or USAJMO, the Junior Olympiad) is based on a composite score known as the USA(J)MO Index. This index combines a student's performance on the AMC (American Mathematics Competitions) and AIME (American Invitational Mathematics Examination).
The formula for the USA(J)MO Index is:
USA(J)MO Index = (AMC 10/12 Score) + 10 * (AIME Score)
This index is crucial for determining which students are invited to take the USAMO or USAJMO.
Components of the Qualification Index
To better understand the qualification index, it's helpful to know how its components are scored:
- AMC 10/12 Score: The AMC 10 and AMC 12 are 25-question multiple-choice exams. Each correct answer typically earns 6 points, no answer earns 1.5 points, and incorrect answers earn 0 points. The maximum possible score is 150 points.
- AIME Score: The AIME is a 15-question short-answer exam. Each correct answer is worth 1 point. Incorrect or unanswered questions are worth 0 points. The maximum possible AIME score is 15 points.
The AIME score is multiplied by 10 in the index calculation to give it significant weight, reflecting the advanced nature of the AIME problems.
Example of USA(J)MO Index Calculation
Let's say a student achieved the following scores:
- AMC 12 Score: 120 points
- AIME Score: 8 points
Using the formula:
USA(J)MO Index = (AMC 12 Score) + 10 * (AIME Score)
USA(J)MO Index = 120 + (10 * 8)
USA(J)MO Index = 120 + 80
USA(J)MO Index = 200
This calculated index is then compared against a cut-off score, which varies annually, to determine eligibility for the USAMO or USAJMO.
Qualification Index Summary
| Component | Description | Points Contribution to Index |
|---|---|---|
| AMC 10/12 Score | Score from the American Mathematics Competition (AMC 10 or AMC 12). Max score: 150 points (6 points/correct, 1.5 points/blank). | Directly added to the index. |
| AIME Score | Score from the American Invitational Mathematics Examination (AIME). Max score: 15 points (1 point/correct). | Multiplied by 10, then added to the index, significantly weighting AIME performance. |
| USA(J)MO Index | (AMC 10/12 Score) + 10 * (AIME Score). This composite score determines eligibility for the USAMO or USAJMO. Cut-off scores for qualification vary each year based on the performance of all participants. Learn more about the AIME. | The final value used for qualification. |
Why the Qualification Index Matters
The USA(J)MO Index serves as the primary gateway to the nation's most prestigious high school mathematics Olympiads. Achieving a high index score signifies exceptional mathematical talent and problem-solving abilities, earning students the invitation to compete at the USAMO or USAJMO.
