# AMC 10/12 Math Contest Preparation

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).

#### Contents

## Contest Information

- 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 \(10^\text{th}\) grade and below, covering high school curriculum up to \(10^\text{th}\) 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).

## Roadmap to Success

We have an entire course focused on the AMC 10/12. Click here if you'd like to just start in on the course.

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.

## Algebra Basics

This includes systems of equations, rates, quadratics, exponents, **floor and ceiling functions**, **fraction part functions**, and logarithms.

\[\left\lfloor \frac { x }{ 1! } \right\rfloor +\left\lfloor \frac { x }{ 2! } \right\rfloor +\left\lfloor \frac { x }{ 3! } \right\rfloor =224\]

Find the integer value of \(x\) that satisfies the equation above.

Note: \(\lfloor x \rfloor\) denotes the greatest integer that is smaller than or equal to \(x\).

## Inequalities

This includes the **Arithmetic Mean-Geometric Mean Inequality** and the **Cauchy-Schwarz Inequality**,

## Polynomials

This includes not only standard school topics like roots and equations but also **Vieta's formulas** and **some helpful transformation tricks**.

## Sequences and Series

This includes arithmetic sequences, geometric sequences, and **telescoping series**.

## Number Theory Basics

Primes numbers, GCD/LCM, and **counting factors** are included.

## Modular Arithmetic

Modular arithmetic includes **systems of congruences** and **Euler's theorem**.

## 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.

\(ABCD\) is a square with \(AB=13\). Points \(E\) and \(F\) are exterior to \(ABCD\) such that \(BE=DF=5\) and \(AE=CF=12\).

If the length of \(EF\) can be represented as \(a\sqrt b, \) where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of any prime, then find \(ab\).

## Analytic Geometry

Coordinate geometry and conics are included here, as well as **mass points** and **complex number geometry**.

## Trigonometry

The standard trig functions are here, as well as the law of cosines, law of sines, trig identities, and **roots of unity**.

\[ \huge \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\hspace{2mm} \frac{\sin x}{\cos x}\hspace{2mm} }{\tan x}\hspace{2mm} }{\cot x}\hspace{2mm} }{\sec x}\hspace{2mm} }{\csc x}\]

Find the value of the above fraction tower if \(0^\circ < x < {90}^\circ\).

## 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**.

In a class of students, \(64\text{%}\) of the students passed the physics exam and \(68\text{%}\) of the students passed the math exam.

What is the minimum percentage of students in the class that passed both exams?

You have three colors of paint, and you want to paint all sides of a regular tetrahedron (a polyhedron with 4 faces that are all equilateral triangles).

How many ways are there to do so if you only paint each face one of your three colors and two colored tetrahedrons are considered equivalent if they can be rotated to look the same?

## Probability

This is a straightforward introduction to probability that includes conditional probability, expected value, **recursion**, **linearity of expectation**, and **events that include states**.

**two** marbles in this bag, **I flipped a coin twice** to determine their colors. For each flip,

- if it was heads \(\rightarrow\) I put in a red marble;
- if it was tails \(\rightarrow\) 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)?

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.

**Cite as:**AMC 10/12 Math Contest Preparation.

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