What Is the AMC Competition?
The AMC, or American Mathematics Competitions, is the first step in the U.S. pathway for selecting mathematics olympiad students. The AMC series includes AMC 8, AMC 10/12, AIME, USAMO, and USAJMO.
AMC 12 is open to students in grade 12 and below. Students who meet the cutoff score receive recognition certificates and are invited to take the American Invitational Mathematics Examination, also known as AIME. The cutoff score changes each year depending on overall performance.
Due to nationality restrictions, Chinese students can generally advance only up to the AIME stage. If U.S. students perform well on AIME, they may qualify for the USA Mathematical Olympiad.
The AMC is not only for “math geniuses.” Any student who is interested in mathematics and enjoys solving problems can participate.
2026 AMC 12 Timeline
The following timeline is for reference.
| Item | AMC 12A | AMC 12B |
|---|---|---|
| Registration Deadline | Expected late October 2026 | Expected early November 2026 |
| Exam Date | Expected early November 2026, Thursday | Expected mid-November 2026, Tuesday |
| Results Release | Around 4–6 weeks after the exam | Around 4–6 weeks after the exam |
What Is the AMC 12 Exam Format?
AMC 12 covers high school mathematics, including trigonometry, advanced algebra, and higher-level algebra. It does not include calculus.
The exam consists of 25 questions in 75 minutes. Students have about 3 minutes per question on average.
| Item | Details |
|---|---|
| Number of Questions | 25 |
| Exam Duration | 75 minutes |
| Average Time per Question | 3 minutes |
| Full Score | 150 points |
AMC 12 Topics Added Beyond AMC 10
| Topic Area | Key Content |
|---|---|
| Advanced Algebra | Complex inequalities, harmonic inequality, cyclic inequality, Cauchy inequality, advanced functions, inverse functions, composite functions, trigonometric identities, sum-to-product and product-to-sum formulas, complex numbers, complex plane, Euler’s formula, De Moivre’s theorem, mathematical induction, advanced sequences and limits |
| Advanced Geometry | Advanced circle geometry, coordinate and geometric integration, two-dimensional and three-dimensional function representations, irregular 2D and 3D figures, 2D vectors and 3D vectors |
| Advanced Number Theory | Quadratic residues, higher-order residues, Fermat’s theorem on sums of two squares, Fermat’s Little Theorem, and various types of Diophantine equations |
| Advanced Combinatorics | Random processes, expected value, and advanced combinatorial problem-solving techniques |
| Integrated Problems | Cross-topic problems combining algebra, geometry, number theory, and combinatorics |
What Is the Difference Between AMC 12A and AMC 12B?
In general, there is no major difference in academic value between AMC 12A and AMC 12B. They are two separate but equally valid exams with similar difficulty levels.
The main difference is the exam date. AMC 12A and AMC 12B are usually held about one week apart. Students may register for either one or both.
Because each paper has different questions, the difficulty may vary slightly from year to year. As a result, cutoff scores may also fluctuate.
Students may choose to take only one paper, but students aiming for AIME qualification or a high AMC 12 score are often advised to take both to increase their chances.
After the contest date, the scoring process usually takes about 3–4 weeks. Students will receive their scores by email.
AMC 12 Sample Problem Analysis
Problem 21
For real numbers (x), let
[
P(x)=1+\cos(x)+i\sin(x)-\cos(2x)-i\sin(2x)+\cos(3x)+i\sin(3x)
]
where (i=\sqrt{-1}).
For how many values of (x) with (0\leq x<2\pi) does (P(x)=0)?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Solution
For (P(x)=0), both the real part and the imaginary part must equal 0.
Therefore:
[
1+\cos x-\cos 2x+\cos 3x=0
]
and
[
\sin x-\sin 2x+\sin 3x=0
]
First consider the imaginary part:
[
\sin x-\sin 2x+\sin(2x+x)=0
]
Using the angle addition formula:
[
\sin x-\sin 2x+\sin 2x\cos x+\cos 2x\sin x=0
]
Rearrange:
[
\sin x(1+\cos 2x)=\sin 2x(1-\cos x)
]
Since
[
\sin 2x=2\sin x\cos x
]
we get:
[
\sin x(1+\cos 2x)=2\sin x\cos x(1-\cos x)
]
If (\sin x=0), then (x=0) or (x=\pi). Substituting both values into the real part equation shows that neither works.
Now divide by (\sin x):
[
1+\cos 2x=2\cos x(1-\cos x)
]
Using
[
\cos 2x=2\cos^2x-1
]
we have:
[
1+2\cos^2x-1=2\cos x(1-\cos x)
]
[
2\cos^2x=2\cos x-2\cos^2x
]
[
4\cos^2x-2\cos x=0
]
[
2\cos x(2\cos x-1)=0
]
So:
[
\cos x=0
]
or
[
\cos x=\frac{1}{2}
]
This gives:
[
x=\frac{\pi}{2},\frac{3\pi}{2}
]
or
[
x=\frac{\pi}{3},\frac{5\pi}{3}
]
Substituting these values into the real part equation shows that none of them satisfy it.
Therefore, there are no values of (x) that make (P(x)=0).
The answer is:
[
\boxed{A}
]
Summer Preparation Strategy for AMC 12 Top 1%
1. Build a Complete Knowledge System
Students should first master the full AMC 12 knowledge framework, including algebra, geometry, number theory, combinatorics, trigonometry, complex numbers, and graph theory.
The goal is to ensure that the first 15 questions can be solved accurately and efficiently. These questions form the foundation of a stable high score.
2. Focus on Advanced Topics
To aim for the top 1%, students must go beyond basic school mathematics.
High-frequency advanced topics include:
| Topic | Key Focus |
|---|---|
| Complex Numbers | Complex plane, polar form, De Moivre’s theorem |
| Number Theory | Modular arithmetic, Fermat’s Little Theorem, Diophantine equations |
| Geometry | Power of a point, circle geometry, analytic geometry |
| Combinatorics | Inclusion-exclusion, expected value, recurrence relations |
| Trigonometry | Trigonometric identities and equation solving |
3. Practice Past Papers Under Timed Conditions
AMC 12 is a time-sensitive exam. Students should complete full past papers under the official 75-minute time limit.
Timed practice helps students improve speed, accuracy, and decision-making. It also trains students to decide which problems to solve, which to skip, and how to manage the final difficult questions.
4. Review Mistakes Systematically
Students should build a personal error log and classify mistakes into different types:
| Error Type | Example |
|---|---|
| Knowledge Gap | Not knowing a theorem or formula |
| Calculation Error | Arithmetic or algebraic mistakes |
| Misreading | Missing key conditions in the problem |
| Strategy Error | Using a slow or inefficient method |
| Time Management Error | Spending too long on one problem |
Regular review is essential for avoiding repeated mistakes and improving score stability.
Final Advice
Summer is the best time to prepare for AMC 12. Students aiming for the top 1% should not rely only on school math or casual practice. They need a structured preparation plan, advanced topic training, timed mock exams, and careful mistake review.
With consistent summer preparation, students can build a stronger foundation, improve problem-solving speed, and increase their chances of qualifying for AIME and achieving a top AMC 12 score.
