Search & glossary

Every rule and concept in AMC 12, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (55)

Advanced counting and probability: Inclusion–exclusion for $n$ sets; derangements $D_n \approx \tfrac{n!}{e}$ ; Catalan $\tfrac{1}{n+1}\binom{2n}{n}$ ; Vandermonde $\sum\binom ak\binom{b}{n-k} = \binom{a+b}{n}$ ; hockey stick $\sum_{i=k}^n\binom ik = \binom{n+1}{k+1}$ .; Formulas & Methods › Geometry, Counting & Probability
AM–GM inequality: For nonnegative reals, $\dfrac{a_1+\cdots+a_n}{n}\ge\sqrt[n]{a_1\cdots a_n}$ , equality when all equal.; Advanced Algebra › Inequalities and Functions
Angle bisectors as an equidistance locus: The bisectors of the angles between $A_1x + B_1y + C_1 = 0$ and $A_2x + B_2y + C_2 = 0$ satisfy $\dfrac{A_1x + B_1y + C_1}{\sqrt{A_1^2+B_1^2}} = \pm\dfrac{A_2x + B_2y + C_2}{\sqrt{A_2^2+B_2^2}}$ — two perpendicular lines, one per sign.; Circles & Lines › The Bisector Between Two Lines
Arithmetic and geometric series: Arithmetic: $S_n = \tfrac n2(a_1+a_n)$ .; Sequences, Series & Recursion › Standard Sums
Change of base — and telescoping logs: $\log_a b = \dfrac{\ln b}{\ln a}$ , so products of logarithms telescope: $\log_a b \cdot \log_b c = \log_a c$ .; Logs & Complex Numbers › Two Powerful Toolkits
Change of base and the reciprocal identity: $\log_b a = \dfrac{\ln a}{\ln b} = \dfrac{1}{\log_a b}$ , and $\log_b a \cdot \log_c b = \log_c a$ .; Logs & Complex Numbers › Logarithm Problems That Look Impossible
Complex numbers: $|z|^2 = z\bar z$ ; $z = re^{i\theta}$ ; de Moivre $(re^{i\theta})^n = r^ne^{in\theta}$ ; $n$ th roots of unity sum to $0$ and are equally spaced; $x^n - 1 = \prod(x - \omega^k)$ ; $|z - a|$ is a distance; multiplication rotates by $\arg$ .; Formulas & Methods › Trigonometry & Complex Numbers
Complex numbers as geometry: Multiplication rotates and scales; $|z - a|$ is a distance; roots of $z^n = r^n$ form a regular $n$ -gon of circumradius $r$ (the roots of $z^6 = 64$ enclose area $6\sqrt3$ ).; Complete Syllabus & Gap Fillers › Gap Fillers
Compute terms until the sequence repeats: For a nonlinear recursion, compute the first several terms and look for a repeat.; Sequences, Series & Recursion › Recursions That Repeat
Conics and coordinates: Circle $(x-h)^2 + (y-k)^2 = r^2$ ; parabola $y^2 = 4px$ (focus $(p, 0)$ ); ellipse $\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1$ , $c^2 = a^2 - b^2$ ; hyperbola $c^2 = a^2 + b^2$ , asymptotes $y = \pm\tfrac bax$ .; Formulas & Methods › Geometry, Counting & Probability
Conics on the AMC 12: Ellipse $\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} = 1$ with $c^2 = a^2 - b^2$ (foci), hyperbola with $c^2 = a^2 + b^2$ , parabola $y^2 = 4px$ with focus $(p, 0)$ .; Complete Syllabus & Gap Fillers › Gap Fillers
De Moivre's theorem: $\left(\cos\theta + i\sin\theta\right)^n = \cos n\theta + i\sin n\theta$ ; in polar form $\left(re^{i\theta}\right)^n = r^ne^{in\theta}$ .; Trigonometry & Complex Numbers › De Moivre, Roots of Unity and Rotation
De Moivre's theorem: $(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta$ — powers of a unit complex number just multiply the angle.; Logs & Complex Numbers › Two Powerful Toolkits
Derivation of change of base: From $x = b^{\log_b x}$ , take $\log_c$ of both sides and pull the exponent down (power rule): $\log_c x = \log_b x \cdot \log_c b$ .; Proofs and Derivations › The Log Rules, Derived
Derivation of the Law of Cosines: Place the triangle with $C$ at the origin and side $a$ along the $x$ -axis: $B = (a, 0)$ and $A = (b\cos C, b\sin C)$ .; Proofs and Derivations › The Trig Identities, Derived
Derivation of the Law of Sines: The area of a triangle is $\tfrac12 ab\sin C$ — computed from ANY vertex.; Proofs and Derivations › The Trig Identities, Derived
Derivation: the formula for $\cos$ of a sum, from rotation: $\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$ .; Proofs and Derivations › Two Derivations the AMC 12 Rewards
Derivation: the sum and product of roots for any degree: Vieta's relations follow from expanding the factored form.; Proofs and Derivations › Two Derivations the AMC 12 Rewards
Double angle from the addition formula: Set $\beta = \alpha$ in $\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ to get $\sin 2\alpha = 2\sin\alpha\cos\alpha$ ; the same move on cosine gives $\cos 2\alpha = \cos^2\alpha - \sin^2\alpha$ .; Proofs and Derivations › The Trig Identities, Derived
Focal definitions do the work: Ellipse: the sum of distances to the two foci is $2a$ .; Circles & Lines › Conics on the AMC 12
Focus and directrix: The parabola $x^2 = 4py$ has focus $(0, p)$ and directrix $y = -p$ ; every point is equidistant from the two.; Circles & Lines › Circles Meet Parabolas
Geometric probability: Probability $=$ favorable area $\div$ total area.; Complete Syllabus & Gap Fillers › Gap Fillers
Geometric series, finite and infinite: $\sum_{k=0}^{n-1} ar^k = a\dfrac{1-r^n}{1-r}$ , and for $|r| < 1$ , $\sum_{k=0}^{\infty} ar^k = \dfrac{a}{1-r}$ .; Sequences, Series & Recursion › Telescoping and Infinite Series
Identities: $\sin^2 + \cos^2 = 1$ ; $\sin(a \pm b)$ , $\cos(a \pm b)$ , $\tan(a \pm b)$ ; double angle $\sin 2a = 2\sin a\cos a$ , $\cos 2a = 2\cos^2a - 1 = 1 - 2\sin^2a$ ; half angle; product-to-sum $2\sin a\cos b = \sin(a+b) + \sin(a-b)$ ; sum-to-product.; Formulas & Methods › Trigonometry & Complex Numbers
Intersecting a circle with a conic: Substitute the parabola into the circle and solve the resulting equation in $u = x^2$ (or $y$ ): each POSITIVE root $u$ contributes two points ( $x = \pm\sqrt u$ ), $u = 0$ one point, and negative roots none.; Circles & Lines › Circles Meet Parabolas
Logarithm laws: $\log_b(xy) = \log_b x+\log_b y$ , $\log_b(x^n) = n\log_b x$ , and $\log_b x = \dfrac{\log x}{\log b}$ .; Advanced Algebra › Logarithms
Logs and exponentials: All log rules; $\log_ab\cdot\log_bc = \log_ac$ ; $a^{\log_ax} = x$ ; $x^{\log_by} = y^{\log_bx}$ .; Formulas & Methods › Algebra & Polynomials
Nearest point of a circle to a line: If the center is at distance $d > r$ from a line, the circle's nearest point to the line is at distance $d - r$ and the farthest at $d + r$ — both along the perpendicular from the center.; Circles & Lines › Circles: Chords, Tangents, Power
Polar form and de Moivre: Write $z = r(\cos\theta+i\sin\theta) = re^{i\theta}$ ; then $z^n = r^n(\cos n\theta + i\sin n\theta)$ .; Trigonometry & Complex Numbers › Complex Numbers
Polynomials in depth: Vieta for any degree; Newton's sums $p_k = e_1p_{k-1} - e_2p_{k-2} + \cdots$ ; remainder theorem $P(a)$ ; factor theorem; conjugate root theorem; rational root theorem; roots-of-unity filter for coefficient sums; Descartes' rule for sign counts.; Formulas & Methods › Algebra & Polynomials
Product and power rules from exponent laws: Let $m = \log_b x$ , $n = \log_b y$ , so $x = b^m$ , $y = b^n$ .; Proofs and Derivations › The Log Rules, Derived
Proof by counting committees: Expanding $(1 + x)^n$ means choosing, from each of the $n$ factors, either the $1$ or the $x$ .; Proofs and Derivations › The Binomial Theorem, Proved
Proof: $\sin^2\theta + \cos^2\theta = 1$: The unit-circle point $(\cos\theta, \sin\theta)$ lies at distance $1$ from the origin, and the distance formula squares to $\cos^2\theta + \sin^2\theta = 1$ — the Pythagorean theorem wearing trig notation.; Proofs and Derivations › The Trig Identities, Derived
Roots of unity: The solutions of $z^n=1$ are $e^{2\pi ik/n}$ , equally spaced on the unit circle, and for $n>1$ they sum to $0$ .; Trigonometry & Complex Numbers › Complex Numbers
Roots of unity sum to zero: The $n$ solutions of $z^n = 1$ are evenly spaced on the unit circle, and their sum is $0$ for $n \ge 2$ .; Logs & Complex Numbers › Two Powerful Toolkits
Sequences, series, recursion: Characteristic equation for linear recurrences; generating functions $\sum a_nx^n$ turn recurrences into algebra; $\sum nx^n = \tfrac{x}{(1-x)^2}$ ; telescoping products; periodic recurrences (compute a cycle).; Formulas & Methods › Algebra & Polynomials
Substitution is the whole method: Feed the equation values that simplify it: $x = 0$ , $x = 1$ , $y = x$ , $y = -x$ , or the value that makes an inner expression collapse.; Advanced Algebra › Functional Equations and Clever Substitution
Symmetry and invariance: If $f(a - x) = f(x)$ the graph is symmetric about $x = \tfrac a2$ ; if $f(x+p) = f(x)$ it is periodic with period $p$ .; Advanced Algebra › Functional Equations and Clever Substitution
Symmetry shortcut: If swapping $x \leftrightarrow y$ (or another visible reflection) exchanges the two lines, then the mirror line of that reflection is automatically one of the bisectors.; Circles & Lines › The Bisector Between Two Lines
Tangency is a distance statement: A line is tangent to a circle exactly when the center's distance to the line EQUALS the radius; it misses when the distance exceeds the radius, and cuts a chord when it is smaller.; Circles & Lines › Distance: Point to Line
Telescoping: If each term is $f(k) - f(k+1)$ , the sum collapses to $f(1) - f(n+1)$ .; Sequences, Series & Recursion › Standard Sums
Telescoping sums: Write each term as a difference: if $t_k = u_k - u_{k+1}$ then $\sum_{k=1}^n t_k = u_1 - u_{n+1}$ .; Sequences, Series & Recursion › Telescoping and Infinite Series
The $a^{\log_a x} = x$ family: $a^{\log_a x} = x$ , $\log_a(a^x) = x$ , and $x^{\log_a y} = y^{\log_a x}$ .; Logs & Complex Numbers › Logarithm Problems That Look Impossible
The checklist: Everything on the AMC 10, plus: trigonometry (identities, sum/product formulas, law of sines/cosines, trig equations); logarithms and exponential equations; complex numbers (de Moivre, roots of unity, modulus geometry); polynomials in depth (Vieta for cubic…; Complete Syllabus & Gap Fillers › What the AMC 12 Tests
The identities you cannot do without: Pythagorean: $\sin^2\theta+\cos^2\theta=1$ .; Trigonometry & Complex Numbers › Trigonometric Identities
The identity, and what it looks like in Pascal's triangle: Claim.; Identities and Techniques, Proved › The Hockey Stick Identity
The one move behind every identity in this topic: Almost every binomial identity a contest asks for is proved the same way: find a set of objects, count it one way, count it another way, and set the two answers equal.; Identities and Techniques, Proved › Counting the Same Thing Twice
The point-to-line distance formula: The distance from $(x_0, y_0)$ to the line $Ax + By + C = 0$ is $d = \dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ ; parallel lines $Ax+By=C_1$ , $Ax+By=C_2$ sit $\dfrac{|C_1 - C_2|}{\sqrt{A^2+B^2}}$ apart.; Circles & Lines › Distance: Point to Line
The three conics in standard form: Ellipse $\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$ with $c^2 = a^2 - b^2$ ; hyperbola with a minus sign and $c^2 = a^2 + b^2$ ; parabola $(x-h)^2 = 4p(y-k)$ with focus $p$ from the vertex.; Circles & Lines › Conics on the AMC 12
The three right-triangle facts: Chord at distance $d$ from the center: length $2\sqrt{r^2 - d^2}$ .; Circles & Lines › Circles: Chords, Tangents, Power
The universal problem-solving loop: Read twice.; Formulas & Methods › Geometry, Counting & Probability
Triangles: Law of sines $\tfrac{a}{\sin A} = 2R$ ; law of cosines; area $\tfrac12ab\sin C$ ; $\text{Area} = \tfrac{abc}{4R} = rs$ .; Formulas & Methods › Trigonometry & Complex Numbers
Two facts do almost everything: If $\omega = e^{2\pi i/n}$ , then (1) $1 + \omega + \omega^2 + \cdots + \omega^{n-1} = 0$ , and (2) $x^n - 1 = (x - 1)(x - \omega)(x - \omega^2)\cdots(x - \omega^{n-1})$ , so $\displaystyle\prod_{k=1}^{n-1}(x - \omega^k) = 1 + x + \cdots + x^{n-1}$ .; Logs & Complex Numbers › Roots of Unity at Work
Two identities that fall out immediately: Set $x = 1$ : $\sum_k \binom{n}{k} = 2^n$ (every subset picks in or out).; Proofs and Derivations › The Binomial Theorem, Proved
Why the squares are the interesting case: The hockey stick adds binomial coefficients.; Identities and Techniques, Proved › Sums of Squares and Cubes, from Binomial Coefficients