AMC, AIME & USAMO Past Problems - Free Math Competition Archive

US Math Competition Pipeline

AMC 8 → AMC 10 / 12 → AIME → USAJMO / USAMO

AMC 8

The AMC 8 is a 25-question, 40-minute, multiple-choice exam for students in grade 8 and below. No calculator is permitted. Problems cover material typically found in grades 6–8 curriculum. Answers viewable on this page

25 Questions 40 Minutes No Calculator Grades 6–8

Year	Problems	Answer Key	Solutions

AMC 10

The AMC 10 is a 25-question, 75-minute, multiple-choice exam for students in grade 10 and below. Covers topics through Algebra II. Two versions (A and B) are offered each year. Top scorers qualify for the AIME. Answers viewable on this page

25 Questions 75 Minutes Versions A & B AIME Qualifier

AIME Qualification: Typically requires a score of ~100–115 on the AMC 10 (varies by year). Top 2.5% of AMC 10 scorers qualify.

Year	Version	Problems	Answer Key	Solutions

AMC 12

The AMC 12 is a 25-question, 75-minute, multiple-choice exam open to all high school students. Covers topics through precalculus. Two versions (A and B) are offered each year. Top scorers qualify for the AIME. Answers viewable on this page

25 Questions 75 Minutes Versions A & B AIME Qualifier

AIME Qualification: Typically requires a score of ~85–100 on the AMC 12 (varies by year). Top 5% of AMC 12 scorers qualify.

Year	Version	Problems	Answer Key	Solutions

AIME (American Invitational Mathematics Examination)

The AIME is a 15-question, 3-hour exam for students who qualify through the AMC 10 or AMC 12. Each answer is an integer from 000 to 999. Two versions (I and II) are offered each year since 2000. Top scorers qualify for the USAJMO or USAMO. Answers viewable on this page

15 Questions 3 Hours Integer Answers (0–999) USAMO Qualifier

USAMO/USAJMO Qualification: Based on AIME score × 10 + AMC score. Cutoffs vary by year.

Year	Version	Problems	Answer Key	Solutions

USAMO & USAJMO

The USA Mathematical Olympiad (USAMO) and USA Junior Mathematical Olympiad (USAJMO) are proof-based competitions for the top ~250–500 AMC/AIME scorers. Each consists of 6 problems over 2 days (3 problems per day, 4.5 hours each day). Solutions require full mathematical proofs.

6 Problems 2 Days × 4.5 Hours Proof-Based ~250–500 Qualifiers

Qualification: Based on a combined AMC + AIME index score. USAJMO is for AMC 10 qualifiers; USAMO is for AMC 12 qualifiers. Top ~250 in each category are invited.

USAMO

Year	Problems	Solutions

USAJMO (since 2010)

Year	Problems	Solutions

Frequently Asked Questions

What is the AMC competition pathway?

The pathway is AMC 8 (middle school) → AMC 10/12 (high school) → AIME (invitational) → USAJMO/USAMO (olympiad). Each level qualifies top scorers for the next. USAMO winners may be selected for the US International Mathematical Olympiad (IMO) team.

What score do I need to qualify for the AIME?

AIME qualification cutoffs vary by year. For AMC 10, typically 100–115 out of 150. For AMC 12, typically 85–100 out of 150. The top ~2.5% of AMC 10 and top ~5% of AMC 12 scorers qualify.

How is the AMC scored?

AMC 10 and AMC 12: 6 points for each correct answer, 1.5 points for each blank, 0 points for each wrong answer (max 150). AMC 8: 1 point per correct answer, no penalty (max 25).

Can I use a calculator on the AMC?

No. Calculators are not permitted on any AMC competition (AMC 8, AMC 10, AMC 12, or AIME). All computations must be done by hand.

What is the difference between USAMO and USAJMO?

Both are proof-based olympiad exams with the same format (6 problems, 2 days). USAJMO is for students who qualified through AMC 10, while USAMO is for AMC 12 qualifiers. USAJMO problems are somewhat easier. USAJMO was introduced in 2010.