Search & glossary

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

Glossary (64)

Addition and double-angle: $\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b$ ; $\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b$ ; $\tan(a+b)=\dfrac{\tan a+\tan b}{1-\tan a\tan b}$ ; $\sin 2a = 2\sin a\cos a$ , $\cos 2a = 1-2\sin^2 a$ .; Trig Formulas That Win › Sum, Double, Triple
Algebra and polynomials: Newton's sums; roots of unity filter; $\prod_{k=1}^{n-1}(1 - \omega^k) = n$ and $\prod\sin\tfrac{k\pi}{n} = \tfrac{n}{2^{n-1}}$ ; Lagrange interpolation; finite differences (degree- $d$ polynomial has constant $d$ th differences); floor: $∑ k/m…; Formulas & Methods › Number Theory & Algebra
Area ratios: same height, same base: Triangles sharing an apex have areas in the ratio of their bases; triangles on the same base have areas in the ratio of their heights.; Cevians & Mass Points › The Ratio-Chasing Recipes
Assign masses so the ratios balance: On a segment split $m:n$ , put mass $n$ at the near end and $m$ at the far end; masses add at intersection points.; Cevians & Mass Points › Mass Points in Practice
Binomial theorem: $(x+y)^n = \sum_{k=0}^n\dbinom nk x^{n-k}y^k$ .; Advanced Counting › Building Recurrences
Ceva's theorem: Cevians $AD$ , $BE$ , $CF$ (with $D, E, F$ on sides $BC$ , $CA$ , $AB$ ) are concurrent if and only if $\dfrac{BD}{DC}\cdot\dfrac{CE}{EA}\cdot\dfrac{AF}{FB} = 1$ .; Geometry: Coordinates & Complex › Cevians: Ceva & Menelaus
Chinese Remainder Theorem: If the moduli are pairwise coprime, a system $x\equiv a_i\pmod{m_i}$ has a unique solution modulo $\prod m_i$ .; Advanced Number Theory › Systems of Congruences
Choosing between them — and mass points: Concurrency of three cevians $\Rightarrow$ Ceva.; Geometry: Coordinates & Complex › Cevians: Ceva & Menelaus
Circle power tools together: Ptolemy's theorem ( $AC \cdot BD = AB \cdot CD + AD \cdot BC$ in a cyclic quadrilateral) and the Power of a Point ( $AP \cdot PB = CP \cdot PD$ for chords through $P$ ) often chain together in one problem.; Telescoping & Recursion › Making Long Computations Short
Condition on the last step: Count by splitting on the final move: a tiling ending in a square leaves $a_{n-1}$ ways, one ending in a domino leaves $a_{n-2}$ , so $a_n = a_{n-1}+a_{n-2}$ .; Advanced Counting › Building Recurrences
Counting: Bijections (Dyck paths, compositions, stars and bars with bounds by PIE); recursion by first choice; generating functions in easy cases; Burnside for symmetric colorings $\tfrac{1}{|G|}\sum|\mathrm{Fix}(g)|$ ; Vandermonde and hockey stick; distributing with …; Formulas & Methods › Counting, Probability & Expectation
Distinct roots: For $a_{n+2}=p\,a_{n+1}+q\,a_n$ , solve $x^2=px+q$ .; Recurrences & Characteristic Equations › The Characteristic Equation Method
Divisor sums are multiplicative: $d(n) = \prod(a_i+1)$ , $\sigma(n) = \prod\dfrac{p_i^{a_i+1}-1}{p_i-1}$ , and both are multiplicative across coprime factors.; Advanced Number Theory › Valuations, Divisor Sums and Digits
Euler's theorem and totient: If $\gcd(a,n)=1$ , then $a^{\varphi(n)}\equiv 1\pmod n$ , where $\varphi(n) = n\prod_{p\mid n}(1-\tfrac1p)$ .; Advanced Number Theory › Orders and Totients
Evaluation tricks: Set $x = 1$ to sum the coefficients; $x = -1$ for the alternating sum; $\tfrac{p(1) + p(-1)}2$ for the even-index coefficients.; Polynomials › Roots, Factors and Clever Evaluations
Extracting every nth coefficient: To sum the coefficients of $x^0, x^n, x^{2n},\dots$ in $P$ , average $P$ over the $n$ th roots of unity: $\tfrac1n\sum_{j} P(\omega^j)$ .; Polynomials › The Roots of Unity Filter
Fibonacci and Binet: Fibonacci is the recurrence $x^2 = x+1$ with roots $\varphi = \tfrac{1+\sqrt5}{2}$ and $\hat\varphi = \tfrac{1-\sqrt5}{2}$ : $F_n = \dfrac{\varphi^n - \hat\varphi^n}{\sqrt5}$ .; Recurrences & Characteristic Equations › The Characteristic Equation Method
Find $F$ with $t_k = F(k + 1) - F(k)$: Useful differences: $k · k!; Telescoping & Recursion › Telescoping Beyond Fractions
Floor functions and base representations: $\lfloor x\rfloor$ problems are block counts: $\sum_{k=1}^{2026}\lfloor\tfrac k7\rfloor$ groups $k$ by quotient; " $n$ divisible by $\lfloor\sqrt n\rfloor$ " gives exactly three per square block.; Complete Syllabus & Gap Fillers › Gap Fillers
Generating functions: Set $F(x) = \sum a_nx^n$ , multiply the recurrence by $x^n$ , sum, and solve for $F$ ; partial fractions recovers $a_n$ .; Recurrences & Characteristic Equations › Solving Recurrences Two Ways
Generating functions, the easy cases: The coefficient of $x^n$ in $(1 + x)^a(1 + x)^b$ is Vandermonde ( $\sum\binom ak\binom{b}{n-k} = \binom{a+b}{n}$ ; $\sum\binom{10}{k}^2 = \binom{20}{10}$ ).; Complete Syllabus & Gap Fillers › Gap Fillers
Geometry: Power of a point; radical axis; Ptolemy; Brahmagupta; Stewart; mass points and Ceva/Menelaus; Heron; $rs = \tfrac{abc}{4R}$ ; Euler $OI^2 = R(R - 2r)$ ; area ratios via shared angles $\tfrac{[ADE]}{[ABC]} = \tfrac{AD\cdot AE}{AB\cdot AC}$ ; coordinates with a …; Formulas & Methods › Geometry & Trigonometry
Iterated functions and periodicity: Digit sums preserve residues mod $9$ and shrink fast — $f(f(f(7^{2026})))$ is a single digit found by size bounds plus mod $9$ .; Complete Syllabus & Gap Fillers › Gap Fillers
Legendre's formula and trailing zeros: $v_p(n!) = \left\lfloor\tfrac np\right\rfloor + \left\lfloor\tfrac n{p^2}\right\rfloor + \cdots$ ; trailing zeros of $n!$ count $v_5(n!)$ . Why it works.; Advanced Number Theory › Valuations, Divisor Sums and Digits
Linear recursions: When each state comes from the previous ones — like $a_n = a_{n-1} + a_{n-2}$ — compute forward from the base cases rather than hunting for a formula.; Telescoping & Recursion › Making Long Computations Short
Linearity of expectation: The expected value of a sum equals the sum of expected values, even when the parts depend on each other.; Advanced Counting › Expected Value
Mass points for ratios ALONG a cevian: Assign masses inversely to the side ratios so each cevian foot balances; then $\dfrac{AP}{PD}=\dfrac{\text{mass}(D)}{\text{mass}(A)}$ at the intersection $P$ .; Cevians & Mass Points › The Ratio-Chasing Recipes
Menelaus, for a line crossing the triangle: For a transversal meeting $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ : $\dfrac{BD}{DC}\cdot\dfrac{CE}{EA}\cdot\dfrac{AF}{FB} = 1$ with signed ratios (the unsigned product is $1$ with an odd number of external points).; Cevians & Mass Points › Mass Points in Practice
Menelaus's theorem: A transversal line crossing lines $BC$ , $CA$ , $AB$ of triangle $ABC$ at $D$ , $E$ , $F$ satisfies $\dfrac{BD}{DC}\cdot\dfrac{CE}{EA}\cdot\dfrac{AF}{FB} = 1$ (unsigned; with signed ratios the product is $-1$ ).; Geometry: Coordinates & Complex › Cevians: Ceva & Menelaus
Newton's identities: Power sums $p_k = \sum r_i^k$ come from the Vieta sums $e_j$ : $p_1 = e_1$ , $p_2 = e_1p_1 - 2e_2$ , $p_3 = e_1p_2 - e_2p_1 + 3e_3$ .; Polynomials › Symmetric Functions of Roots
Number theory: CRT; orders and primitive roots; $\varphi$ , $\tau$ , $\sigma$ multiplicative; Legendre and Kummer; LTE $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ ; Wilson; quadratic residues ( $-1$ QR iff $p \equiv 1 \bmod 4$ ); base- $b$ digit sums $\equiv n \pmod{b-1}$ ; $(a^m…; Formulas & Methods › Number Theory & Algebra
Order of an element: The order of $a$ mod $n$ is the least $d>0$ with $a^d\equiv 1$ , and it always divides $\varphi(n)$ .; Advanced Number Theory › Orders and Totients
Pairing tricks for long sums: Pair to a constant: $\sin^2 k^\circ + \sin^2(90-k)^\circ = 1$ ; $\tan k^\circ\tan(90-k)^\circ = 1$ ; $(1+\tan k^\circ)(1+\tan(45-k)^\circ) = 2$ .; Trig Formulas That Win › Products & Sums That Collapse
Partial fractions first: A rational term with factored denominator splits into pieces whose difference telescopes: $\dfrac{1}{k(k+2)} = \tfrac12\left(\dfrac1k - \dfrac1{k+2}\right)$ .; Telescoping & Recursion › Finding the Difference That Telescopes
Pick the tool by what is given: Given many lengths and a right angle: coordinates.; Geometry: Coordinates & Complex › Lengths by Coordinates, Trig and Power of a Point
Probability and expectation: Linearity with indicators; expected waiting times by first-step equations; Markov-chain states for "until two in a row"; geometric probability (unit square regions, broken stick); conditional probability by restricting the sample space; symmetry arguments f…; Formulas & Methods › Counting, Probability & Expectation
Product-to-sum and sum-to-product: $2\sin A\sin B = \cos(A-B) - \cos(A+B)$ ; $\sin A + \sin B = 2\sin\tfrac{A+B}2\cos\tfrac{A-B}2$ .; Trig Formulas That Win › Sums That Collapse
Proof of Ptolemy's theorem: In cyclic $ABCD$ , mark $E$ on diagonal $BD$ with $\angle BCE = \angle ACD$ .; Heavy Machinery, Proved › Ptolemy's Theorem, Proved
Proof of Stewart's theorem: Let cevian $AD = d$ hit $BC$ with $BD = m$ , $DC = n$ .; Heavy Machinery, Proved › Stewart's Theorem and the Bisector, Proved
Proof: the arithmetic mean–geometric mean inequality for two terms: $\dfrac{a+b}{2} \ge \sqrt{ab}$ for $a, b \ge 0$ , with equality iff $a = b$ .; Heavy Machinery, Proved › Two More Proofs Worth Carrying
Proof: the distances from one vertex multiply to $n$: Divide $x^n - 1$ by $x - 1$ : $\prod_{k=1}^{n-1}(x - \omega_k) = 1 + x + \cdots + x^{n-1}$ .; Heavy Machinery, Proved › Roots of Unity, Proved
Proof: the law of cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ .; Heavy Machinery, Proved › Two More Proofs Worth Carrying
Reading equality: when Ptolemy becomes an inequality: For a NON-cyclic quadrilateral the same construction gives $AC \cdot BD < AB \cdot CD + AD \cdot BC$ — so equality is a TEST for being cyclic.; Heavy Machinery, Proved › Ptolemy's Theorem, Proved
Repeated roots: If the characteristic equation has a double root $r$ , the general solution gains a factor of $n$ : $a_n = (A + Bn)\,r^n$ .; Recurrences & Characteristic Equations › The Characteristic Equation Method
Rotations by multiplication: Rotating a point $z$ about the origin by $\theta$ is multiplication by $e^{i\theta}$ ; about a center $c$ , use $c + (z-c)e^{i\theta}$ .; Geometry: Coordinates & Complex › Complex Numbers as Points
Set up states, then solve the linear system: Name $p_A$ , $p_B$ , … for the probability of the goal from each state; write one equation per state from its transitions; solve.; Advanced Counting › Probability With States and Symmetry
Shift, flip, or reduce: For $a_n = c\,a_{n-1} + d$ with $c \ne 1$ , subtract the fixed point $L = \tfrac{d}{1 - c}$ : then $a_n - L = c(a_{n-1} - L)$ , a geometric sequence.; Recurrences & Characteristic Equations › Constant Terms, Substitutions and Periodicity
Shoelace formula: A polygon with vertices $(x_i,y_i)$ in order has area $\tfrac12\left|\sum_i (x_iy_{i+1} - x_{i+1}y_i)\right|$ .; Geometry: Coordinates & Complex › Analytic Geometry
Symmetry collapses the work: If the labels can be permuted without changing the process, the corresponding probabilities are equal — solve one and multiply.; Advanced Counting › Probability With States and Symmetry
Telescoping products: A product telescopes when each factor is a ratio of consecutive values: $\prod_{k=2}^{n}\dfrac{k-1}{k} = \dfrac1n$ .; Telescoping & Recursion › Finding the Difference That Telescopes
Telescoping sums: Partial fractions turn $\dfrac{1}{k(k+1)}$ into $\dfrac1k - \dfrac{1}{k+1}$ , so the sum telescopes: $\sum_{k=1}^{n} \dfrac{1}{k(k+1)} = 1 - \dfrac{1}{n+1}$ .; Telescoping & Recursion › Making Long Computations Short
Telescoping with cotangents and tangents: $\arctan a - \arctan b = \arctan\dfrac{a-b}{1+ab}$ , and $\cot\theta - \cot2\theta = \csc2\theta$ .; Trig Formulas That Win › Sums That Collapse
The 60-degree product identity: $\sin\theta\,\sin(60^\circ-\theta)\,\sin(60^\circ+\theta)=\dfrac{\sin 3\theta}{4}$ , and the same shape holds for cosine and tangent ( $\tan$ version multiplies to $\tan 3\theta$ ).; Trig Formulas That Win › Products & Sums That Collapse
The angle bisector length, derived from Stewart: A bisector from $A$ splits $BC$ in ratio $c : b$ (bisector ratio theorem), so $m = \tfrac{ac}{b+c}$ , $n = \tfrac{ab}{b+c}$ .; Heavy Machinery, Proved › Stewart's Theorem and the Bisector, Proved
The characteristic equation: For $a_n = pa_{n-1} + qa_{n-2}$ , solve $r^2 = pr + q$ : distinct roots give $a_n = Ar_1^n + Br_2^n$ , a repeated root gives $a_n = (A + Bn)r^n$ .; Recurrences & Characteristic Equations › Solving Recurrences Two Ways
The checklist: Integer answers $000$ – $999$ , fifteen problems, three hours.; Complete Syllabus & Gap Fillers › What the AIME Tests
The factor and remainder theorems, used backwards: $p(a) = 0 \iff (x-a) \mid p$ ; the remainder of $p$ on division by $x - a$ is $p(a)$ .; Polynomials › Roots, Factors and Clever Evaluations
The law of cosines as a length machine: $c^2 = a^2 + b^2 - 2ab\cos C$ ; with $\cos C$ known from another triangle, one unknown length falls out.; Geometry: Coordinates & Complex › Lengths by Coordinates, Trig and Power of a Point
The universal problem-solving loop: Read twice.; Formulas & Methods › Geometry & Trigonometry
Trigonometry: Product-to-sum and sum-to-product; $\tan$ addition; $\sum_{k=1}^{n-1}\cos\tfrac{2k\pi}{n} = -1$ ; $\cos\tfrac\pi7\cos\tfrac{2\pi}7\cos\tfrac{3\pi}7 = \tfrac18$ (doubling trick); telescoping via $\tan(a) - \tan(b) = \tfrac{\sin(a - b)}{\cos a\cos b}$ ; $ x…; Formulas & Methods › Geometry & Trigonometry
Triple angle in disguise: $\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ and $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$ .; Trig Formulas That Win › Sum, Double, Triple
Turn every length ratio into an area ratio: Two triangles with the same apex and bases on one line have areas in the ratio of their bases.; Cevians & Mass Points › Areas from Ratios
Which theorem for which picture: Three cevians through ONE point: Ceva, $\frac{BD}{DC}\cdot\frac{CE}{EA}\cdot\frac{AF}{FB}=1$ .; Cevians & Mass Points › The Ratio-Chasing Recipes
Why $x^n - 1$ factors over the $n$ th roots of unity: By De Moivre (proved by induction on the angle-addition formula), the $n$ numbers $\omega_k = \text{cis}\tfrac{2\pi k}{n}$ all satisfy $x^n = 1$ ; a degree- $n$ polynomial has at most $n$ roots, so these are ALL of them and $x^n - 1 = \prod_k (x - \omega_k)$ .; Heavy Machinery, Proved › Roots of Unity, Proved