Formula Sheet

AMC 8 / 10 / 12 formulas, strategies, and pattern tricks. Click any formula to copy its LaTeX.

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Algebra#Practice →16

Difference of squares8+: $a^2-b^2=(a-b)(a+b)$
Perfect squares8+: $(a\pm b)^2=a^2\pm 2ab+b^2$
Sum/difference of cubes10+: $a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$
Quadratic formula10+: $ax^2+bx+c=0 \Rightarrow x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$ ; discriminant $\Delta=b^2-4ac$
Vertex of a parabola10+: $y=ax^2+bx+c$ has vertex at $x=-\dfrac{b}{2a}$
Distance, rate, time8+: $d=rt$ ; average speed $=\dfrac{\text{total distance}}{\text{total time}}$
Exponent & log rules10+: $a^m a^n=a^{m+n}$ , $(a^m)^n=a^{mn}$ , $\log_b(xy)=\log_b x+\log_b y$ , $\log_b x=\dfrac{\log x}{\log b}$
Vieta's formulas10+: For $ax^2+bx+c$ : $r_1+r_2=-\frac{b}{a}$ , $r_1r_2=\frac{c}{a}$
Simon's Favorite Factoring Trick10+: $xy+ax+by=c \Rightarrow (x+b)(y+a)=c+ab$ ↗ 2008 AMC 12B #16
Work backwards / plug in choices8+: Test the choices, starting with the middle one (C) to see whether to go up or down
Symmetric sums10+: $x^2+y^2=(x+y)^2-2xy$ ; $\ x^3+y^3=(x+y)^3-3xy(x+y)$
Vieta (cubic)12+: $x^3+bx^2+cx+d$ (monic): $\sum r=-b$ , $\sum r_ir_j=c$ , $r_1r_2r_3=-d$
AM–GM10+: For $a,b\ge0$ : $\dfrac{a+b}{2}\ge\sqrt{ab}$ , equality iff $a=b$
Complex numbers12+: $i^2=-1$ ; $|a+bi|=\sqrt{a^2+b^2}$ ; $e^{i\theta}=\cos\theta+i\sin\theta$ (De Moivre: $(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$ )
$x+\frac1x=k$ 10+: Square it: $x^2+\frac1{x^2}=k^2-2$ (uses )
Nested radical $\sqrt{a+\sqrt{a+\cdots}}$ 10+: Set it equal to $x$ : $x=\sqrt{a+x}$ , so $x^2=a+x$ (uses )

Geometry#Practice →21

Pythagorean theorem & triples8+: $a^2+b^2=c^2$ . Triples: $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $8$ - $15$ - $17$ , $7$ - $24$ - $25$ (and multiples)
Special right triangles8+: $45$ - $45$ - $90$ : $1:1:\sqrt2$ . $\quad 30$ - $60$ - $90$ : $1:\sqrt3:2$
Areas8+: Triangle $\frac12bh$ ; trapezoid $\frac12(b_1+b_2)h$ ; circle $\pi r^2$ ; circumference $2\pi r$
Angles in a polygon8+: Interior sum $=180^\circ(n-2)$ ; exterior angles sum to $360^\circ$
Equilateral triangle10+: Area $=\dfrac{\sqrt3}{4}s^2$ , height $=\dfrac{\sqrt3}{2}s$
Volumes8+: Prism/cylinder $Bh$ ; pyramid/cone $\frac13Bh$ ; sphere $\frac43\pi r^3$ , surface $4\pi r^2$
Area with an angle10+: $A=\tfrac12ab\sin C$ (two sides and the included angle)
Similar figures10+: Lengths scale by $k$ $\Rightarrow$ areas by $k^2$ , volumes by $k^3$
Heron's formula10+: $A=\sqrt{s(s-a)(s-b)(s-c)}$ , $s=\frac{a+b+c}{2}$
Power of a point10+: Chords: $AP\cdot PB=CP\cdot PD$ . Secant–tangent: $PT^2=PA\cdot PB$
Tangent lines10+: Radius $\perp$ tangent at the point of contact; two tangents from one point are equal
Angle bisector theorem10+: Bisector from $A$ splits $BC$ so that $\dfrac{BD}{DC}=\dfrac{AB}{AC}$
Mass points10+: Assign weights to vertices so each cevian balances: $m_B\cdot BD=m_C\cdot DC$ ↗ 2004 AMC 10B #20 ↗ 2016 AMC 10A #19
Coordinates10+: Place a figure on axes (right angle at origin). Distance $\sqrt{(\Delta x)^2+(\Delta y)^2}$ ; Shoelace: $A=\frac12\left|\sum(x_iy_{i+1}-x_{i+1}y_i)\right|$
Inradius & circumradius10+: $A=rs$ ; $\ A=\dfrac{abc}{4R}$ . Right triangle: $R=\frac{c}{2}$ , $r=\frac{a+b-c}{2}$
Law of sines / cosines12+: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}=2R$ ; $\ c^2=a^2+b^2-2ab\cos C$
Pick's theorem10+: Lattice polygon: $A=I+\frac{B}{2}-1$ ( $I$ interior, $B$ boundary points)
Ptolemy's theorem12+: Cyclic quadrilateral $ABCD$ : $AC\cdot BD=AB\cdot CD+AD\cdot BC$
Inscribed angle10+: Inscribed angle $=\frac12$ its arc; angle in a semicircle is $90^\circ$ ; opposite angles of a cyclic quadrilateral sum to $180^\circ$
$13$ - $14$ - $15$ triangle10+: Altitude to side $14$ is $12$ , splitting it into $5$ - $12$ - $13$ and $9$ - $12$ - $15$ ; area $=84$ (uses , )
$120^\circ$ or $60^\circ$ angle10+: Drop a perpendicular to make $30$ - $60$ - $90$ triangles, or use law of cosines (uses , )

Number Theory#Practice →13

Divisibility rules8+: $3,9$ : digit sum; $4$ : last two digits; $8$ : last three; $11$ : alternating digit sum; $5,10,2$ : last digit
GCD × LCM8+: $\gcd(a,b)\cdot\operatorname{lcm}(a,b)=ab$
Number of divisors8+: If $n=p_1^{a_1}\cdots p_k^{a_k}$ : $d(n)=(a_1+1)\cdots(a_k+1)$
Sum of divisors10+: $\sigma(n)=\prod\left(1+p_i+\cdots+p_i^{a_i}\right)$
Modular arithmetic10+: If $a\equiv b$ and $c\equiv d \pmod m$ then $a+c\equiv b+d$ , $ac\equiv bd\pmod m$
Units digits cycle10+: Powers' last digits repeat (e.g. $2\to4\to8\to6$ ). Reduce the exponent mod the cycle length.
Trailing zeros of $n!$ 10+: $\left\lfloor\frac n5\right\rfloor+\left\lfloor\frac n{25}\right\rfloor+\left\lfloor\frac n{125}\right\rfloor+\cdots$ (Legendre's formula with $p=5$ )
Euclidean algorithm10+: $\gcd(a,b)=\gcd(b,\,a \bmod b)$ , e.g. $\gcd(2n+1,n+3)=\gcd(n+3,5)$
Base conversion10+: $(d_k\ldots d_1d_0)_b=\sum d_ib^i$ ; convert to base 10, compute, convert back
Fermat's little theorem10+: $p$ prime, $p\nmid a$ : $a^{p-1}\equiv1\pmod p$
Euler's theorem12+: $\gcd(a,n)=1\Rightarrow a^{\varphi(n)}\equiv1\pmod n$ , $\ \varphi(n)=n\prod_{p\mid n}\left(1-\frac1p\right)$
Chinese Remainder Theorem12+: Congruences mod pairwise coprime $m_i$ have a unique solution mod $\prod m_i$
Is it a perfect square?8+: Squares are $\equiv0,1\pmod4$ and end in $0,1,4,5,6,9$ (uses )

Counting & Probability#Practice →13

Permutations & combinations8+: $P(n,k)=\dfrac{n!}{(n-k)!}$ , $\ \binom nk=\dfrac{n!}{k!(n-k)!}$
Arrangements with repeats8+: $\dfrac{n!}{a!\,b!\cdots}$ (e.g. MISSISSIPPI: $\frac{11!}{4!4!2!}$ )
Probability8+: $P=\dfrac{\text{favorable}}{\text{total}}$ (equally likely outcomes); independent: $P(A\cap B)=P(A)P(B)$
Inclusion–exclusion10+: $|A\cup B|=|A|+|B|-|A\cap B|$ ; three sets add back $|A\cap B\cap C|$
Expected value10+: $E[X]=\sum x\,P(x)$ ; linearity: $E[X+Y]=E[X]+E[Y]$ always
Complementary counting10+: Count what you don't want and subtract from the total
Stars and bars10+: Nonnegative solutions of $x_1+\cdots+x_k=n$ : $\binom{n+k-1}{k-1}$ ; positive: $\binom{n-1}{k-1}$
Casework8+: Split by a key feature (first digit, number of a color). Make cases non-overlapping.
Circular arrangements10+: $n$ people around a table: $(n-1)!$ (divide by $2$ more if flips count as the same)
Geometric probability10+: Probability $=\dfrac{\text{favorable area}}{\text{total area}}$ when choosing points uniformly
Pascal & binomial theorem10+: $\binom nk=\binom{n-1}{k-1}+\binom{n-1}{k}$ ; $(x+y)^n=\sum\binom nk x^{n-k}y^k$ ; $\sum_k\binom nk=2^n$
Hockey stick identity12+: $\sum_{i=r}^{n}\binom ir=\binom{n+1}{r+1}$
Derangements12+: $D_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}$ ; $D_1..D_5=0,1,2,9,44$

Sequences & Series#Practice →5

Arithmetic sequence8+: $a_n=a_1+(n-1)d$ ; $\ S_n=\dfrac{n(a_1+a_n)}{2}$
Geometric sequence10+: $a_n=a_1r^{n-1}$ ; $\ S_n=a_1\dfrac{1-r^n}{1-r}$ ; infinite ( $|r|<1$ ): $\dfrac{a_1}{1-r}$
Classic sums8+: $1+\cdots+n=\frac{n(n+1)}2$ ; $\ \sum k^2=\frac{n(n+1)(2n+1)}6$ ; $\ \sum k^3=\left(\frac{n(n+1)}2\right)^2$ ; first $n$ odds $=n^2$
Telescoping10+: $\dfrac{1}{k(k+1)}=\dfrac1k-\dfrac1{k+1}$ , so most terms cancel
Find the pattern10+: Compute the first several terms; look for a cycle or formula, then extend

Test Strategy#6

Scoring (AMC 10/12)8+: $6$ per correct, $1.5$ per blank, $0$ per wrong; max $150$ . AMC 8: $1$ per correct, no penalty — never leave a blank.
Pacing8+: AMC 10/12: 25 questions in 75 minutes (~3 min each). AMC 8: 25 in 40 minutes. Skip and return.
Estimate & eliminate8+: Draw diagrams to scale and eyeball lengths/areas to rule out choices.
Small cases8+: Replace $2026$ with $1,2,3$ to find a pattern.
Answer form clues10+: A $\sqrt3$ in choices hints at $30$ - $60$ - $90$ or equilateral triangles; $\pi$ hints at circles.
Guess or leave blank?10+: A random guess is worth $1.2$ on average vs. $1.5$ for a blank; guess only after eliminating at least $2$ choices