AMC 10/12 Strategy & Reference Cheat Sheet

Key Facts

• The AMC 10 and AMC 12 each have $25$ multiple-choice questions in $75$ minutes, so average pacing is $3$ minutes per problem.
• The scoring rule is $6$ points for a correct answer, $1.5$ points for a blank answer, and $0$ points for an incorrect answer.
• Use the identity $a^2-b^2=(a-b)(a+b)$ to factor differences of squares quickly.
• For a quadratic $ax^2+bx+c=0$, the roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and their sum is $-\frac{b}{a}$.
• If integers are counted modulo $n$, then $a\equiv b\pmod{n}$ means $n$ divides $a-b$.
• The area of a triangle can be found with $A=\frac{1}{2}bh$ or $A=\frac{1}{2}ab\sin C$.
• For independent events, $P(A\cap B)=P(A)P(B)$, and for complementary events, $P(A)=1-P(A^c)$.
• When order matters, use permutations such as $P(n,r)=\frac{n!}{(n-r)!}$, and when order does not matter, use combinations such as $\binom{n}{r}=\frac{n!}{r!(n-r)!}$.

Vocabulary

Modular arithmetic

A system for comparing integers by their remainders after division by a fixed positive integer.

Invariant

A quantity or property that stays the same while other parts of a problem change.

Complement

The set of outcomes in a probability problem that are not in the event being considered.

Combination

A selection of objects where order does not matter, often counted by $\binom{n}{r}$.

Similar triangles

Triangles with equal corresponding angles and proportional corresponding side lengths.

Vieta's formulas

Rules that connect the roots of a polynomial to its coefficients, such as the root sum $-\frac{b}{a}$ for $ax^2+bx+c=0$.