The AMC (American Mathematics Competition) 10 and 12 are math contests in the United States and should not be confused with AMC (Australian Mathematics Competition).
The AMC 10 and 12 are intended for high school level students. They are 25-question, 75-minute, multiple choice tests with an emphasis on problem-solving.
The AMC 10 specifically is for students in 10th grade and below, covering high school curriculum up to 10th grade.
The AMC 12 includes trigonometry, advanced algebra, and advanced geometry, but excludes calculus.
Students who do well on the AMC 10 or AMC 12 are invited to participate in the AIME, and a high score on AIME can lead to qualification for the United States of America Mathematics Olympiad (USAMO).
This document is meant to guide you through our content that's worthwhile for contest study, not just from the course, but other portions of the site. There are sample problems to each of the topics covered (so you diagnose where you might need more work).
Topics that are not commonly taught in school textbooks are marked in bold. If you've already mastered your school classes, these are the topics you won't necessarily know yet. (Your experiences may vary! Try the sample problems if you're not sure.)
Note that the last chapter of the AMC course is on general strategies, but it's recommended you don't save it for last.
Triangle > Rectangle > Circle
Circle > Rectangle > Triangle
Rectangle > Triangle > Circle
Circle > Triangle > Rectangle
Rectangle > Circle > Triangle
Triangle > Circle > Rectangle
The diagram shows how a mobile will be balanced when left to hang.
What are the relative weights of these shapes?
If positive numbers x, y, and z satisfy x+y+z=75, what is the maximum value of
x+5y+7z?
Polynomials
This includes not only standard school topics like roots and equations but also Vieta's formulas and some helpful transformation tricks.
How many of the fractions above cannot be reduced?
Find the tens digit of 201420142014.
Synthetic Geometry
The basics here are as follows: Pythagorean theorem, triangle areas, similar triangles, angle bisector theorem, power of a point, cyclic quadrilaterals, and circles.
Find the value of the above fraction tower if 0∘<x<90∘.
How many of the 14th roots of unity lie on the axes of the complex plane?
Combinatorics
Straightforward use of the rule of product is in this section as well as complementary counting, binomial coefficients, inclusion-exclusion and ball and urn problems.
As a tourist in NY, I want to go from the Grand Central Station (42nd street and 4th avenue) to Times Square (47th street and 7th avenue).
If I only walk West and North, how many ways are there for me to get there?
2 percent32 percent34 percent37 percent
In a class of students, 64% of the students passed the physics exam and 68% of the students passed the math exam.
What is the minimum percentage of students in the class that passed both exams?
15
27
30
81
You have three colors of paint, and you want to paint all sides of a regular tetrahedron (a polyhedron with 4 equilateral triangles faces).
How many ways are there to do so if you only paint each face one of your three colors?
(Rotations of the tetrahedron are considered equivalent).
(71+1)71−(71−1)71
What is the last digit of the number above?
Probability
This is a straightforward introduction to probability that includes conditional probability, expected value, recursion, linearity of expectation, and events that include states.
When I put two marbles in this bag, I flipped a coin twice to determine their colors. For each flip,
if it was heads → I put in a red marble;
if it was tails → I put in a blue marble.
You reach into my bag and randomly take out one of the two marbles. It is red. You put it back in. Then you reach into the bag again. What is the chance that, this time, you pull out a blue marble?
There are 3 envelopes, one with $1, another with $2, and the other with $4. You do not know which envelope has which amount of money.
After you pick one at random, you are told which envelope of the remaining 2 has the lowest value, but not the actual value of that envelope. What is the expected value if you switch to the last envelope remaining (the one that has not yet been mentioned)?
36118161112
None of the above
Two fair dice are rolled, and it is revealed that (at least) one of the numbers rolled was a 4. What is the probability that the other number rolled was a 6?
Note: You are not told which of the numbers rolled is a 4.