Formulas & Facts Cheat Sheet

Items marked (must) should be instant recall. Everything here appears repeatedly in AMC 10 problems 1–20.

Algebra

• (must) Vieta for $ax^2+bx+c$: roots sum to $-b/a$, multiply to $c/a$. Discriminant $b^2-4ac$; a perfect square means rational roots.
• (must) $(a+b)^2 = a^2+2ab+b^2$, $a^2-b^2=(a-b)(a+b)$, $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$, $(a+b+c)^2 = a^2+b^2+c^2+2(ab+bc+ca)$.
• Simon's Favorite Factoring Trick: $xy + ax + by = (x+b)(y+a) - ab$.
• $a^2+b^2 = (a+b)^2 - 2ab = (a-b)^2 + 2ab$; $\frac{1}{a}+\frac{1}{b} = \frac{a+b}{ab}$.
• (must) Arithmetic series: $\frac{n(\text{first}+\text{last})}{2}$; $1+2+\dots+n = \frac{n(n+1)}{2}$; $1^2+\dots+n^2 = \frac{n(n+1)(2n+1)}{6}$.
• (must) Geometric series: $a\frac{r^n-1}{r-1}$; infinite $\frac{a}{1-r}$ for $|r|<1$.
• Telescoping: $\frac{1}{k(k+1)} = \frac{1}{k}-\frac{1}{k+1}$.
• Rate problems: work $=$ rate $\times$ time; combined rates add; average speed $=$ total distance / total time (never the average of speeds).
• Percent change compounds multiplicatively: up 20% then down 20% is $\times 0.96$.
• AM-GM: $\frac{a+b}{2} \ge \sqrt{ab}$, equality iff $a=b$; product fixed → sum minimized when equal.
• Absolute value: $|x-a| < r$ means $a-r < x < a+r$; $|x|$ graphs are V-shapes; split into cases at the zeros.
• Exponents: $a^m a^n = a^{m+n}$, $(a^m)^n = a^{mn}$, $a^{-n} = 1/a^n$; $\sqrt{a}\sqrt{b} = \sqrt{ab}$; $\log_b x = y \iff b^y = x$.

Number theory

• (must) Number of divisors of $p_1^{e_1}\cdots p_k^{e_k}$ is $(e_1+1)\cdots(e_k+1)$; sum of divisors is $\prod \frac{p_i^{e_i+1}-1}{p_i-1}$.
• (must) $\gcd(a,b)\cdot\text{lcm}(a,b) = ab$. Euclidean algorithm: $\gcd(a,b) = \gcd(b, a \bmod b)$.
• (must) Divisibility rules: 2, 4, 8 (last 1, 2, 3 digits); 3, 9 (digit sum); 5, 10; 6 (2 and 3); 11 (alternating digit sum); 7 (double last digit, subtract from the rest).
• Primes under 100: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 (25 of them). To test $n < 121$ for primality, check divisibility by 2, 3, 5, 7 only.
• Consecutive integers: product of $k$ consecutive integers is divisible by $k!$. A perfect square has an odd number of divisors and is $\equiv 0$ or $1 \pmod 4$.
• Last digits cycle: powers of 2 → 2,4,8,6; of 3 → 3,9,7,1; of 7 → 7,9,3,1; of 4 → 4,6; of 9 → 9,1. $n^5 \equiv n \pmod{10}$.
• Repeating decimals: $0.\overline{ab} = \frac{ab}{99}$; $\frac{1}{7} = 0.\overline{142857}$. A fraction in lowest terms terminates iff the denominator has only factors 2 and 5.
• Base $b$: the digits of $n$ are the remainders on repeated division by $b$. Digit sum $\equiv n \pmod{b-1}$.
• Squares to memorize: $11^2=121,\ 12^2=144,\ 13^2=169,\ 14^2=196,\ 15^2=225,\ 16^2=256,\ 17^2=289,\ 18^2=324,\ 19^2=361,\ 21^2=441,\ 23^2=529,\ 25^2=625$. Powers of 2 up to $2^{12}=4096$. $2^{10} = 1024$.

Counting & probability

• (must) $n!$, $P(n,k) = \frac{n!}{(n-k)!}$, $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, $\binom{n}{k}=\binom{n}{n-k}$, $\sum_k \binom{n}{k} = 2^n$.
• (must) Arrangements with repeats: $\frac{n!}{a!\,b!\,\cdots}$. Circular arrangements of $n$: $(n-1)!$. Necklaces (flips allowed): $(n-1)!/2$.
• (must) Stars and bars: nonnegative solutions of $x_1+\dots+x_k = n$ number $\binom{n+k-1}{k-1}$; positive solutions $\binom{n-1}{k-1}$.
• (must) Inclusion-exclusion: $|A\cup B| = |A|+|B|-|A\cap B|$; three sets add back the triple intersection.
• Complementary counting: "at least one" $=$ total $-$ "none". Adjacent objects: glue them into a block.
• Lattice paths from $(0,0)$ to $(m,n)$ with right/up steps: $\binom{m+n}{m}$.
• Probability: $P = \frac{\text{favorable}}{\text{total}}$ when outcomes are equally likely; independent events multiply; $P(A \text{ or } B) = P(A)+P(B)-P(A\text{ and }B)$.
• Expected value is linear: $E[X+Y] = E[X]+E[Y]$ even when dependent. Expected number of successes in $n$ trials with probability $p$ is $np$.
• Two dice: 36 outcomes; sum $s$ has $6-|s-7|$ ways. Geometric probability: favorable region / total region.
• Handshakes among $n$ people: $\binom{n}{2}$. Diagonals of an $n$-gon: $\frac{n(n-3)}{2}$.

Geometry

• (must) Triangle area: $\frac{1}{2}bh = \frac{1}{2}ab\sin C = rs = \sqrt{s(s-a)(s-b)(s-c)}$ (Heron, $s$ = semiperimeter). Equilateral side $a$: area $\frac{\sqrt3}{4}a^2$, height $\frac{\sqrt3}{2}a$.
• (must) Special right triangles: $45$-$45$-$90$ sides $1:1:\sqrt2$; $30$-$60$-$90$ sides $1:\sqrt3:2$. Pythagorean triples: (3,4,5), (5,12,13), (8,15,17), (7,24,25), (9,40,41), (20,21,29) and multiples.
• (must) Similar triangles: corresponding sides in ratio $k$, areas in ratio $k^2$; parallel line through a triangle cuts off a similar triangle.
• Triangle inequality: $a+b>c$. Angle bisector theorem: the bisector from $A$ divides $BC$ in ratio $AB:AC$. Medians meet at the centroid, which divides each median $2:1$.
• Right triangle altitude to the hypotenuse: $h^2 = pq$, $a^2 = pc$ (segments $p, q$ of the hypotenuse $c$). Inradius of a right triangle: $r = \frac{a+b-c}{2}$.
• (must) Circles: circumference $2\pi r$, area $\pi r^2$, arc length $\frac{\theta}{360}\cdot 2\pi r$, sector area $\frac{\theta}{360}\pi r^2$. Inscribed angle is half the central angle; angle in a semicircle is right; tangent $\perp$ radius; two tangents from a point are equal.
• Power of a point: $PA\cdot PB = PC \cdot PD$ for chords through $P$; tangent-secant $PT^2 = PA\cdot PB$.
• Polygon angles: sum $(n-2)\cdot 180^\circ$; each exterior angle of a regular $n$-gon is $360^\circ/n$. Regular hexagon of side $a$: area $\frac{3\sqrt3}{2}a^2$.
• Trapezoid area $\frac{(b_1+b_2)}{2}h$; rhombus/kite area $\frac{d_1 d_2}{2}$. Shoelace for $(x_1,y_1),\dots$: $\frac12|\sum (x_i y_{i+1} - x_{i+1} y_i)|$.
• Coordinates: distance $\sqrt{(\Delta x)^2+(\Delta y)^2}$; midpoint averages; perpendicular slopes multiply to $-1$; distance from $(x_0,y_0)$ to $ax+by+c=0$ is $\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}$.
• Solids: prism/cylinder volume $=$ base area $\times$ height; pyramid/cone $\frac13$ of that; sphere $V=\frac43\pi r^3$, $S = 4\pi r^2$; cube diagonal $a\sqrt3$; cone slant height $\sqrt{r^2+h^2}$, lateral area $\pi r \ell$.
• Reflections: shortest path that touches a line → reflect the endpoint across the line and draw a straight segment.

Statistics, logic, time

• Mean $=$ sum / count, so "mean is an integer" is a divisibility condition. Median of an even count averages the two middle values. Mode must appear most often (unique mode: strictly more).
• Adding a value $v$ to a set of $n$ numbers with mean $m$ changes the mean by $\frac{v-m}{n+1}$.
• Clock hands: the minute hand moves $6^\circ$ per minute, the hour hand $0.5^\circ$; they overlap every $\frac{720}{11}$ minutes. Angle at $h{:}m$ is $|30h - 5.5m|$.
• Calendar: 365 days $= 52$ weeks $+ 1$ day, so the same date moves one weekday forward per year (two after a leap day). Leap years: divisible by 4, except centuries not divisible by 400.