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

Glossary (153)

AM–GM and Cauchy–Schwarz: AM–GM: $\dfrac{\sum a_i}{n}\ge\sqrt[n]{\prod a_i}$ .; Inequalities › The Essential Inequalities
AM–GM under a product or sum constraint: With $abc = 1$ : $a + b + c + \tfrac1a + \tfrac1b + \tfrac1c \ge 6$ (six terms with product $1$ ).; Olympiad Inequalities II: The Full Ladder › Every Standard Inequality, Placed
Bertrand's postulate: For every $n \ge 1$ the interval $(n, 2n]$ contains a prime; for $n > 3$ there is one strictly between $n$ and $2n - 2$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Multiplicative functions and Möbius
Bounding: squeeze between consecutive powers: If $m^2 < N < (m + 1)^2$ then $N$ is not a square.; Olympiad Number Theory IV: Primitive Roots, Residues & Bounding › Quadratic Residues and Size Arguments
Break the problem into claims: Before solving, write the two or three intermediate facts that would make the problem easy ("Lemma 1: $f$ is injective.; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Lemma Generation
Building polynomials from symmetric data: Given $r + s + t$ , $rs + st + tr$ , $rst$ , the monic cubic with those roots is $x^3 - e_1x^2 + e_2x - e_3$ .; Olympiad Algebra II: Polynomials, Recurrences & Radical Equations › Algebra at Proof Level
Cauchy and its pathologies: $f(x+y) = f(x) + f(y)$ forces $f(qx) = qf(x)$ for rational $q$ , hence $f(x) = cx$ on $\mathbb{Q}$ .; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Functional equations: the pointwise trap
Cauchy–Schwarz, Titu and Hölder: For real $a_i,b_i$ : $\left(\sum a_ib_i\right)^2\le\left(\sum a_i^2\right)\left(\sum b_i^2\right)$ , with equality exactly when $(a_i)$ and $(b_i)$ are proportional.; Inequalities › Cauchy–Schwarz: Choosing What to Pair
Cauchy–Schwarz: Engel form and the geometric form: Engel (Titu): $\sum\tfrac{a_i^2}{b_i} \ge \tfrac{(\sum a_i)^2}{\sum b_i}$ .; Olympiad Inequalities II: The Full Ladder › Every Standard Inequality, Placed
Ceva and Menelaus for free: Cevians through $(0 : v : w)$ , $(u : 0 : w)$ , $(u : v : 0)$ concur iff the product of ratios is $1$ — Ceva is the determinant of the three lines vanishing.; Olympiad Geometry IV: Barycentric & Complex Coordinates › Barycentric Coordinates
Check the equality case first: Before hunting bounds, ask WHERE equality would hold and whether the constraint permits it — an unattainable equality point means the true extremum sits at a boundary instead.; Inequalities › Good-to-Know Inequalities
Choose the modulus from the orders: In an equation such as $a^x\pm b^y=c^z$ , reduce modulo an $m$ for which the powers involved have short periods — the period of $a^x\bmod m$ is $\operatorname{ord}_m(a)$ .; Olympiad Number Theory III: Exponential Diophantine Equations › Exponential Equations: Choosing the Modulus
Choosing the bash: barycentric, complex, or Cartesian: Barycentric suits triangle centres, cevians and conics through the vertices; complex numbers suit circles (put the circumcircle on $|z| = 1$ : then $h = a+b+c$ , and the foot of a perpendicular has a one-line formula); Cartesian suits right angles, midpoints,…; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Directed angles and the bash menu
Collinearity, concyclicity, perpendicularity: $a, b, c$ collinear iff $\frac{a - b}{a - c}$ is real, i.e.; Olympiad Geometry IV: Barycentric & Complex Coordinates › Complex Numbers in Geometry
Combinatorial geometry: Finite point sets in the plane: convex hull arguments (the extreme points), the extremal principle (closest pair, smallest triangle), pigeonhole on regions (divide the square into $n$ pieces), and colorings/parity of lattice points.; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Games, Monovariants and Combinatorial Geometry
Combinatorics: Pigeonhole (all forms); extremal principle; invariants and monovariants; colorings and weightings; double counting; bijections; induction on structures; PIE; generating functions; Catalan and reflection; Erdős–Szekeres; graph theory (degrees, trees, biparti…; Formulas & Methods › Number Theory & Combinatorics
Coordinate lemmas that save an hour: British flag: for a point $P$ and rectangle $ABCD$ , $PA^2 + PC^2 = PB^2 + PD^2$ .; Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals › The Circle Toolkit
Coordinates relative to the triangle: A point is $P = (u : v : w)$ with $u + v + w = 1$ , where $u = \frac{[PBC]}{[ABC]}$ etc.; Olympiad Geometry IV: Barycentric & Complex Coordinates › Barycentric Coordinates
Cross ratio: For four collinear points, $(A,B;C,D) = \dfrac{\overline{AC}}{\overline{BC}} : \dfrac{\overline{AD}}{\overline{BD}}$ ; for four concyclic points, use the complex formula $\dfrac{(a-c)(b-d)}{(a-d)(b-c)}$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Projective Tools
Cyclic quadrilaterals: Ptolemy and Brahmagupta: A quadrilateral is cyclic iff opposite angles sum to $180^\circ$ (or iff an angle equals the exterior angle at the opposite vertex).; Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals › The Circle Toolkit
Difference of squares and difference of powers: $m^2 - n^2 = 2^k$ splits as $(m-n)(m+n)$ with both factors powers of $2$ of the same parity; $2^n + 1 = m^2$ becomes $2^n = (m-1)(m+1)$ , two powers of $2$ differing by $2$ , so $n = 3$ ; $3^x - 2^y = 1$ needs $x$ even (mod $8$ ) and then $(3^x/2 - 1)(3^x/2…; Olympiad Number Theory III: Exponential Diophantine Equations › Equations in Exponents
Difference of squares and factorization counts: $x^2 - y^2 = N$ has positive solutions in bijection with factorizations $N = de$ , $d < e$ , $d \equiv e \pmod 2$ .; Olympiad Number Theory II: Residues, Diophantine Equations & Factorials › Deeper Arithmetic
Direct, contradiction, contrapositive: To prove $P\Rightarrow Q$ : argue directly; or assume $P$ and $\lnot Q$ and reach a contradiction; or prove the contrapositive $\lnot Q\Rightarrow\lnot P$ .; Proof Techniques › The Main Forms
Directed angles mod $180^\circ$: Write $\angle(\ell, m)$ for the angle rotating line $\ell$ to line $m$ , taken modulo $180^\circ$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Directed angles and the bash menu
Divide out the smallest power, then look at parity: $2^x + 2^y = 2^z$ : dividing by $2^{\min(x,y)}$ leaves $1 + 2^k = 2^m$ , forcing $k = 0$ ; so the solutions are exactly $(x, x, x+1)$ .; Olympiad Number Theory III: Exponential Diophantine Equations › Equations in Exponents
Double counting: Count the same set of pairs (or triples) two ways.; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Graph Theory and Double Counting
Double counting and graphs: Count incidences two ways to derive an identity or inequality (the handshake lemma is the prototype).; Olympiad Combinatorics › Existence and Impossibility
Extremal boards: bound by partition, achieve by pattern: To maximize black cells on a $6\times6$ board with no fully black $2\times2$ : partition into nine $2\times2$ blocks, each holding at most $3$ — bound $27$ ; achieve it by whitening every cell with both coordinates even.; Olympiad Combinatorics II: Colorings, Games & Graphs › Structures and Strategies
Extremal principle: Consider the largest, smallest, or closest object; its extremal property forces structure.; Pigeonhole & Extremal › Two Existence Principles
Extremal principle and pigeonhole: Look at the largest, smallest, or closest object — extremeness forces structure.; Olympiad Combinatorics › Existence and Impossibility
Factorials: gaps in trailing zeros and Euler's theorem: $v_5(n!)$ jumps by $2$ at multiples of $25$ and by $3$ at multiples of $125$ , so some zero-counts ( $5, 11, 17, \ldots, 29, 30$ ) never occur.; Olympiad Number Theory II: Residues, Diophantine Equations & Factorials › Deeper Arithmetic
Fibonacci diagonals and subset conditions: Subsets of $\{1..n\}$ whose size equals their minimum number $F_n$ : sum $\binom{n-k}{k-1}$ over $k$ , a shallow diagonal of Pascal's triangle.; Bijections, Homothety & Structured Counting › Counting by Correspondence
Find the linear recurrence hiding inside: If $a_{n+1}a_{n-1}=a_n^2+c$ with all $a_n\ne0$ , then $\dfrac{a_{n+1}+a_{n-1}}{a_n}$ is the same for every $n$ ; calling it $k$ , \[a_n+1=k a_n-a_n-1.\] If $k$ and the first two terms are integers, every term is an integer.; Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences › Integer-Valued Recurrences
Forbidden phrases: "Clearly", "obviously", "it is easy to see", "similarly" (unless the symmetry is genuine and stated), "by a well-known theorem" (name it), "WLOG" (say what symmetry justifies it).; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Rigorous Exposition
Functional equations: Substitute strategically ( $x = y = 0$ , $y = -x$ , $y = 1$ ), look for injectivity or surjectivity, and guess the form ( $f(x + y) = f(x) + f(y) + xy$ suggests $\tfrac{x^2}{2} + cx$ ).; Olympiad Algebra & Functional Equations › Proof-Level Algebra
Games: strategy stealing, pairing, and P/N positions: To show the first player wins without finding the strategy: strategy stealing (an extra move never hurts).; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Games, Monovariants and Combinatorial Geometry
Games: work backward from the end: A position is losing (P) iff every move leads to a winning (N) position; label backward from the terminal state.; Olympiad Combinatorics II: Colorings, Games & Graphs › Structures and Strategies
Graphs: degrees, trees, bipartite, Turán: The degree sum is $2|E|$ (so the number of odd-degree vertices is even).; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Graph Theory and Double Counting
Growth arguments and $a^b = b^a$: $a^b = b^a$ with $a < b$ has the unique solution $(2, 4)$ : $\tfrac{\ln t}{t}$ increases on $(0, e)$ and decreases after, so one of the two is $\le 2$ .; Olympiad Number Theory III: Exponential Diophantine Equations › Equations in Exponents
Guess the structure, then prove it: Once small cases suggest "the answer is $n$ even" or " $f$ must be linear", the proof splits into two halves: show the guessed objects work (usually easy) and show nothing else does (the real problem).; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Scratchpad Strategy
Hall's marriage theorem: A bipartite graph with parts $X$ , $Y$ has a matching saturating $X$ iff every $S\subseteq X$ satisfies $|N(S)| \ge |S|$ .; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Graphs: connectivity, extremal edges, planarity
Harmonic bundles: $(A,B;C,D) = -1$ is called harmonic.; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Projective Tools
Helly's theorem: If finitely many convex sets in the plane have the property that every three of them share a point, then all of them share a point.; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Set systems and the probabilistic method
Homothety in figures: The medial triangle is the image of $ABC$ under the homothety at the centroid with ratio $-\tfrac12$ ; the anticomplementary triangle (lines through each vertex parallel to the opposite side) has ratio $-2$ and four times the area.; Olympiad Geometry › Homothety and Spiral Similarity
Homothety, spiral similarity, inversion: Homothety maps a figure to a scaled copy (tangent circles, midpoints, the nine-point circle).; Olympiad Geometry › The Configurations That Recur
How the practice tests work: The USAMO practice sets draw six problems at a time from the hand-verified USAMO pool (number theory, combinatorics, geometry, algebra, inequalities, games).; USAMO Practice Tests & Write-Up Playbook › Sit a Practice Test
Incircle lengths: Tangent lengths $s - a, s - b, s - c$ ; $r = \tfrac{K}{s}$ ; $AI = \tfrac{r}{\sin(A/2)}$ , so $AI^2 = (s-a)^2 + r^2$ ; the $A$ -excircle has radius $r_a = \tfrac{K}{s - a}$ and tangent length $s$ from $A$ .; Olympiad Geometry III: Circumcircle, Power of a Point, Incircle › Circles Doing the Work
Induction replaces continuity: On $\mathbb N$ , $\mathbb Z$ or $\mathbb Q$ no regularity hypothesis is needed: find $f$ on the generators and propagate.; Functional Equations With Tricky Conditions › Functional Equations on $\mathbb N$ , $\mathbb Z$ and $\mathbb Q$
Induction, weak and strong: Prove a base case, then show it for $n$ (weak) — or for all values below $n$ (strong) — implies it for $n+1$ .; Proof Techniques › The Main Forms
Inequalities: AM–GM (weighted too); Cauchy–Schwarz and Engel; QM–AM–GM–HM; power means; Jensen; rearrangement and Chebyshev; Schur; SOS; Hölder $\left(\sum a_i^3\right)\left(\sum b_i^3\right)\left(\sum c_i^3\right) \ge \left(\sum a_ib_ic_i\right)^3$ ; Muirhead for symmetr…; Formulas & Methods › Algebra & Inequalities
Injectivity, surjectivity, and iteration: From $f(f(n)) = n + 4$ : $f$ is injective (compose the hypothesis), and applying $f$ once more gives $f(n + 4) = f(n) + 4$ , so $f$ is determined by four values.; Functional Equations With Tricky Conditions › Pinning Down the Function
Integer polynomials: For $P \in \mathbb Z[x]$ : $a - b \mid P(a) - P(b)$ .; Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences › Polynomials and Sequences at Proof Level
Interpolation with a forced zero: If a polynomial of degree $\le n$ agrees with a known pattern at $n$ points, subtract the pattern: the difference has those points as roots, so it is $c\prod(x - x_i)$ and one extra condition pins $c$ .; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Symmetric functions and power sums
Invariant: Find a quantity — a parity, a sum mod $k$ , a coloring — that a process never changes.; Invariants & Monovariants › Process Problems
Invariants: An invariant is a quantity a process never changes.; Invariants & Monovariants › Proof Ideas That Win
Invariants and monovariants: To prove a process can never reach a state, find a quantity it preserves (parity, a sum mod $m$ , a coloring count) that differs between start and goal.; Olympiad Combinatorics › Existence and Impossibility
Isogonal conjugates and the symmedian: Reflecting $AP$ , $BP$ , $CP$ in the bisectors of $\angle A$ , $\angle B$ , $\angle C$ gives three lines that concur at the isogonal conjugate $P^{*}$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Directed angles and the bash menu
Jensen's inequality: For convex $f$ and weights $w_i \ge 0$ summing to $1$ : $f\left(\sum w_ix_i\right) \le \sum w_if(x_i)$ ; reversed for concave $f$ .; Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences › Jensen and Rearrangement
Legendre's formula: $v_p(n!) = \sum_{k\ge1}\left\lfloor\dfrac{n}{p^k}\right\rfloor = \dfrac{n - s_p(n)}{p-1}$ , where $s_p(n)$ is the digit sum of $n$ in base $p$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Valuations, residues and reciprocity
Lifting the exponent: For odd prime $p \mid a - b$ , $p \nmid ab$ : $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ ; for $p \mid a + b$ and odd $n$ : $v_p(a^n + b^n) = v_p(a + b) + v_p(n)$ .; Olympiad Number Theory › The Proof-Level Toolkit
Lifting the exponent: For odd prime $p \mid a - b$ with $p \nmid ab$ : $v_p(a^n - b^n) = v_p(a - b) + v_p(n)$ .; Olympiad Number Theory IV: Primitive Roots, Residues & Bounding › Orders and Primitive Roots
Linear recurrences and their characteristic roots: $a_{n+2} = pa_{n+1} + qa_n$ has general term $A\alpha^n + B\beta^n$ where $\alpha, \beta$ solve $r^2 = pr + q$ (with $(A + Bn)\alpha^n$ for a double root).; Olympiad Algebra II: Polynomials, Recurrences & Radical Equations › Algebra at Proof Level
Mantel and Turán: Triangle-free on $n$ vertices: at most $\left\lfloor n^2/4\right\rfloor$ edges (Mantel), attained by the balanced complete bipartite graph.; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Graphs: connectivity, extremal edges, planarity
Matchings, gaps, and forced adjacency: Perfect matchings of $2n$ people: $(2n - 1)!!; Olympiad Combinatorics II: Colorings, Games & Graphs › Structures and Strategies
Mixtilinear incircles: The $A$ -mixtilinear incircle is tangent to rays $AB$ , $AC$ and internally tangent to the circumcircle at $T$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Named Configurations
Möbius inversion: $\sum_{d\mid n}\mu(d) = [\,n = 1\,]$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Multiplicative functions and Möbius
Möbius substitutions and cyclic systems: If $g(x) = \tfrac{1}{1-x}$ then $g^3 = \mathrm{id}$ : an equation $f(x) + f(g(x)) = x$ yields three linear equations in $f(x), f(gx), f(g^2x)$ , solved by adding and subtracting.; Functional Equations With Tricky Conditions › Pinning Down the Function
Monovariant: A quantity that only increases (or only decreases) each step proves the process must terminate.; Invariants & Monovariants › Process Problems
Monovariants for termination: A quantity that strictly decreases (or increases) and is bounded proves the process ends.; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Games, Monovariants and Combinatorial Geometry
Newton's identities: With $p_k = \sum r_i^k$ : $p_1 = e_1$ , $p_2 = e_1p_1 - 2e_2$ , $p_3 = e_1p_2 - e_2p_1 + 3e_3$ , and in general $p_k = e_1p_{k-1} - e_2p_{k-2} + \cdots + (-1)^{k-1}ke_k$ .; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Symmetric functions and power sums
Number theory: Orders, primitive roots, CRT; Fermat, Euler, Wilson; LTE (all cases); Legendre, Kummer; quadratic residues, Euler's criterion, reciprocity and both supplements; Vieta jumping; infinite descent; Zsigmondy (citable); $\gcd(a^m - 1, a^n - 1)$ ; sums over diviso…; Formulas & Methods › Number Theory & Combinatorics
Orders and primitive roots: The order of $a$ mod $n$ divides $\varphi(n)$ and divides any $k$ with $a^k \equiv 1$ .; Olympiad Number Theory › The Proof-Level Toolkit
Pell equations: $x^2 - dy^2 = 1$ with $d$ a positive non-square has infinitely many solutions; the fundamental one comes from the continued fraction of $\sqrt d$ , and all others are $(x_1 + y_1\sqrt d)^n$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Valuations, residues and reciprocity
Pigeonhole principle: If $n+1$ objects go into $n$ boxes, some box holds at least two; more generally some box holds at least $\lceil m/n\rceil$ of $m$ objects.; Pigeonhole & Extremal › Two Existence Principles
Planarity and Euler: For a connected planar graph, $v - e + f = 2$ .; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Graphs: connectivity, extremal edges, planarity
Poles and polars: For a circle of centre $O$ and radius $r$ , the polar of $P$ is the line perpendicular to $OP$ through the inverse point $Q$ , where $OP\cdot OQ = r^2$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Projective Tools
Polynomials and functional equations: Vieta and Newton; $a - b \mid P(a) - P(b)$ ; rational root theorem; Eisenstein and mod- $p$ irreducibility; Lagrange interpolation and finite differences; roots of unity filter; Chebyshev polynomials for boundedness.; Formulas & Methods › Algebra & Inequalities
Polynomials at proof level: Integer polynomials: $a - b \mid P(a) - P(b)$ .; Olympiad Algebra & Functional Equations › Proof-Level Algebra
Power of a point and radical axes: For a point $P$ and circle, $PA\cdot PB$ is constant over lines through $P$ (negative inside).; Olympiad Geometry › The Configurations That Recur
Power of a point, three ways: For a point $P$ and a circle with center $O$ , radius $r$ : every line through $P$ meeting the circle at $X, Y$ has $PX\cdot PY = |PO^2 - r^2|$ — equal to the tangent length squared when $P$ is outside.; Olympiad Geometry III: Circumcircle, Power of a Point, Incircle › Circles Doing the Work
Proof by picture is not a proof: A diagram supports the exposition and must not carry any logical weight.; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Rigorous Exposition
Proof: $\sum_{d\mid n}\mu(d) = [\,n = 1\,]$: The Möbius function sums to zero over the divisors of any $n > 1$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Proofs & Why It Matters
Proof: $p_3 = e_1p_2 - e_2p_1 + 3e_3$ for three variables: Newton's identity in the case the course uses most.; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Proofs & Why It Matters
Proof: a forest on $n$ vertices with $c$ components has $n - c$ edges: Hence any graph with at least $n$ edges contains a cycle.; Olympiad Combinatorics II: Colorings, Games & Graphs › Proofs & Why It Matters
Proof: additivity glues the pointwise choice: If $f$ is additive and $f(x)^2 = x^2$ for all $x$ , then $f = \operatorname{id}$ or $f = -\operatorname{id}$ .; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Proofs & Why It Matters
Proof: Bouton's theorem for Nim: A Nim position is losing for the player to move iff the XOR of heap sizes is $0$ .; Olympiad Combinatorics II: Colorings, Games & Graphs › Proofs & Why It Matters
Proof: Euler's criterion: For odd prime $p$ and $p \nmid a$ : $a^{(p-1)/2} \equiv 1$ if $a$ is a QR, $\equiv -1$ otherwise.; Olympiad Number Theory II: Residues, Diophantine Equations & Factorials › Proofs & Why It Matters
Proof: Legendre's formula: $v_p(n!) = \sum_{k\ge1}\left\lfloor n/p^k\right\rfloor$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Proofs & Why It Matters
Proof: LYM, hence Sperner: For an antichain $\mathcal F$ of subsets of $[n]$ , $\sum_{A\in\mathcal F}\binom{n}{|A|}^{-1}\le 1$ .; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Proofs & Why It Matters
Proof: Mantel's theorem: A triangle-free graph on $n$ vertices has at most $\left\lfloor n^2/4\right\rfloor$ edges.; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Proofs & Why It Matters
Proof: Ptolemy's theorem: For cyclic $ABCD$ : $AC\cdot BD = AB\cdot CD + AD\cdot BC$ .; Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals › Proofs & Why It Matters
Proof: Schur's inequality: For nonnegative $a, b, c$ : $a(a - b)(a - c) + b(b - a)(b - c) + c(c - a)(c - b) \ge 0$ .; Olympiad Inequalities II: The Full Ladder › Proofs & Why It Matters
Proof: tangent lengths $BD = s - b$: The incircle contact points split the sides into the tangent lengths $s - a$ , $s - b$ , $s - c$ .; Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals › Proofs & Why It Matters
Proof: the characteristic-root formula for two-term recurrences: If $\alpha \ne \beta$ solve $r^2 = pr + q$ , every solution of $a_{n+2} = pa_{n+1} + qa_n$ is $A\alpha^n + B\beta^n$ .; Olympiad Algebra II: Polynomials, Recurrences & Radical Equations › Proofs & Why It Matters
Proof: the rational root theorem: If $\tfrac pq$ in lowest terms is a root of $a_nx^n + \cdots + a_0$ with integer coefficients, then $p \mid a_0$ and $q \mid a_n$ .; Olympiad Algebra II: Polynomials, Recurrences & Radical Equations › Proofs & Why It Matters
Proof: the rearrangement inequality: For $a_1 \le \cdots \le a_n$ and $b_1 \le \cdots \le b_n$ , the sum $\sum a_ib_{\sigma(i)}$ is maximized by $\sigma = \text{id}$ and minimized by the reversing permutation.; Olympiad Inequalities II: The Full Ladder › Proofs & Why It Matters
Proof: the Simson criterion: The feet of the perpendiculars from $P$ to the sides of $ABC$ are collinear iff $P$ is on the circumcircle.; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Proofs & Why It Matters
Proof: the symmedian ratio $BX/XC = c^2/b^2$: The reflection of the median in the bisector cuts $BC$ in the ratio of the squares of the adjacent sides.; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Proofs & Why It Matters
Proof: Wilson's theorem: $(p - 1)!; Olympiad Number Theory II: Residues, Diophantine Equations & Factorials › Proofs & Why It Matters
Quadratic residues and the two supplements: For an odd prime $p$ : $-1$ is a QR iff $p \equiv 1 \pmod 4$ ; $2$ is a QR iff $p \equiv \pm1 \pmod 8$ .; Olympiad Number Theory II: Residues, Diophantine Equations & Factorials › Deeper Arithmetic
Radical and rational equations: the ghost problem: Squaring, clearing denominators, and cross-multiplying can INTRODUCE solutions.; Olympiad Algebra II: Polynomials, Recurrences & Radical Equations › Algebra at Proof Level
Rearrangement and Chebyshev: For sequences sorted the same way, $\sum a_ib_i$ is maximized when paired in the same order and minimized when paired oppositely.; Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences › Jensen and Rearrangement
Rearrangement and smoothing: A sum $\sum a_ib_i$ is largest when both sequences are sorted the same way; "smoothing" nudges variables toward equality while keeping the constraint.; Inequalities › The Essential Inequalities
Residues that rule out solutions: $-1$ is a square mod $p$ iff $p \equiv 1 \pmod 4$ ; $2$ is a square iff $p \equiv \pm1 \pmod 8$ ; $-3$ iff $p \equiv 1 \pmod 3$ .; Olympiad Number Theory IV: Primitive Roots, Residues & Bounding › Quadratic Residues and Size Arguments
Rooks, permutations, and forbidden positions: Non-attacking rook placements on an $n\times n$ board are permutations; forbidding the main diagonal gives derangements $D_n = n!\sum_{k\le n}(-1)^k/k!$ , and forbidding neighbors around a circle of $2n$ people gives a matching count by inclusion–exclusion o…; Bijections, Homothety & Structured Counting › Counting by Correspondence
Schur, SOS, and smoothing: Schur ( $t=1$ ), for $a,b,c\ge0$ : $\sum a(a-b)(a-c)\ge0$ , i.e.; Olympiad Inequalities II: The Full Ladder › Every Standard Inequality, Placed
Sequences: periodicity, growth, and telescoping: For recurrences, find the invariant (a conserved quantity like $a_{n+1}a_{n-1} - a_n^2$ ), the closed form (characteristic roots, or a trigonometric substitution when $a_{n+1} = 2a_n^2 - 1$ ), or the monotone bounded structure.; Olympiad Algebra III: Jensen, Rearrangement, Polynomials & Sequences › Polynomials and Sequences at Proof Level
Set up, compute, stop: A coordinate solution has three parts, each of which must be written: (1) the setup — which coordinates, where the origin is, which quantities are free; (2) the formulas for every point the problem defines, in the order they are defined; (3) the target cond…; Olympiad Geometry IV: Barycentric & Complex Coordinates › Bashes Worked Start to Finish
Simson line: The feet of the perpendiculars from $P$ to the three sides of triangle $ABC$ are collinear exactly when $P$ lies on the circumcircle.; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Named Configurations
Small cases, a table, a guess, two directions: Compute the smallest cases by hand and put them in a table.; Everything USAMO Tests › What the First Hour Looks Like
Sperner, antichains and LYM: An antichain of subsets of an $n$ -set has at most $\binom{n}{\lfloor n/2\rfloor}$ members (Sperner).; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Set systems and the probabilistic method
Structure the reader can follow: State what you will prove.; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Rigorous Exposition
Substitute to isolate, then verify: $f(x^2 + y) = f(x)^2 + y$ : replace $y$ by $y - x^2$ to get $f(y) = y + c$ , then substitute back to force $c = 0$ .; Functional Equations With Tricky Conditions › Pinning Down the Function
Synthetic and analytic geometry: Angle chasing and cyclic quadrilaterals (opposite angles, equal inscribed angles, the exterior-angle criterion); power of a point and radical axes/center; Ptolemy (and its inequality); Brahmagupta; Ceva and Menelaus (trig forms too); Stewart, Apollonius, an…; Formulas & Methods › Geometry
Tangent lengths and the incircle: From an external point the two tangent segments to a circle are equal.; Olympiad Geometry II: Circles, Tangents & Cyclic Quadrilaterals › The Circle Toolkit
The arc midpoint and the incenter–excenter lemma: If the bisector from $A$ meets the circumcircle again at $M$ , then $MB = MC = MI = MI_A$ : $M$ is the center of the circle through $B$ , $C$ , the incenter $I$ , and the $A$ -excenter $I_A$ .; Olympiad Geometry III: Circumcircle, Power of a Point, Incircle › Circles Doing the Work
The difference lemma: If $P$ has integer coefficients and $a\ne b$ are integers, then $a-b\mid P(a)-P(b)$ .; Olympiad Algebra & Functional Equations › Integer Polynomials: $a-b\mid P(a)-P(b)$
The elementary symmetric polynomials: For roots $r_1,\dots,r_n$ : $e_1 = \sum r_i$ , $e_2 = \sum_{i<j} r_ir_j$ , …, $e_n = \prod r_i$ , and $\prod(x - r_i) = x^n - e_1x^{n-1} + e_2x^{n-2} - \cdots$ .; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Symmetric functions and power sums
The extremal principle: Look at the largest, smallest, or closest element — its extremeness forces structure the middle elements do not have.; Invariants & Monovariants › Proof Ideas That Win
The first ninety minutes: Read all three problems.; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Scratchpad Strategy
The gaps graders look for: USAMO Practice Tests & Write-Up Playbook › Finding the Gap in Your Own Proof
The incenter/excenter lemma ("Fact 5"): Let $M$ be the midpoint of arc $BC$ not containing $A$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Named Configurations
The inequality ladder: AM–GM (with equality iff all equal: $\tfrac ab + \tfrac bc + \tfrac ca \ge 3$ ), Cauchy–Schwarz and its Engel form $\sum\tfrac{a_i^2}{b_i} \ge \tfrac{(\sum a_i)^2}{\sum b_i}$ , rearrangement, Chebyshev, Jensen for convex functions, Schur, and the SOS (sum of …; Olympiad Algebra & Functional Equations › Proof-Level Algebra
The Legendre symbol and reciprocity: $\left(\tfrac{a}{p}\right)$ is $+1$ if $a$ is a nonzero square mod $p$ , $-1$ otherwise.; Olympiad Number Theory V: Legendre, Pell & Möbius › Valuations, residues and reciprocity
The order controls everything: If $a^n \equiv 1 \pmod p$ then $\text{ord}_p(a) \mid n$ , and $\text{ord}_p(a) \mid p - 1$ .; Olympiad Number Theory IV: Primitive Roots, Residues & Bounding › Orders and Primitive Roots
The pigeonhole principle: Placing more than $n$ objects into $n$ boxes forces some box to hold at least two.; Invariants & Monovariants › Proof Ideas That Win
The probabilistic method: To prove an object exists, put a probability distribution on candidates and show the bad event has probability less than $1$ , or that the expected value of some count is small enough that some outcome beats the average.; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Set systems and the probabilistic method
The proof-writing method: Restate the claim; fix notation; prove lemmas before use; name every theorem; cover every case (including degenerate ones); state the equality condition; conclude.; Formulas & Methods › Geometry
The QM–AM–GM–HM chain: For positive reals: $\sqrt{\dfrac{\sum a_i^2}{n}}\ \ge\ \dfrac{\sum a_i}{n}\ \ge\ \sqrt[n]{\prod a_i}\ \ge\ \dfrac{n}{\sum \tfrac{1}{a_i}}$ , all with equality exactly when the numbers are equal.; Inequalities › Good-to-Know Inequalities
The reflection bijection: Paths from $(0,0)$ to $(n,n)$ that never go above $y = x$ number $\binom{2n}{n} - \binom{2n}{n-1} = C_n$ : reflect each bad path across $y = x + 1$ after its first touch to get a bijection with all paths to $(n-1, n+1)$ .; Bijections, Homothety & Structured Counting › Counting by Correspondence
The roots of unity filter: Let $\omega=e^{2\pi i/n}$ .; Olympiad Algebra & Functional Equations › Roots of Unity: Filters and Factor Tests
The standard multiplicative functions: $\tau(n)$ (number of divisors), $\sigma(n)$ (their sum), $\varphi(n)$ (Euler totient) and $\mu(n)$ (Möbius: $0$ if $n$ has a square factor, else $(-1)^{\#\text{primes}}$ ) are all multiplicative: $f(mn) = f(m)f(n)$ for coprime $m, n$ .; Olympiad Number Theory V: Legendre, Pell & Möbius › Multiplicative functions and Möbius
The substitution playbook: Try $x = y = 0$ ; $y = x$ ; $y = -x$ ; $x = 0$ alone; swapping $x$ and $y$ to exploit symmetry; and substituting a value that makes one side constant.; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Functional equations: the pointwise trap
The trap: Deducing " $f(x) = x$ or $f(x) = -x$ for each $x$ " does NOT give " $f = \operatorname{id}$ or $f = -\operatorname{id}$ ".; Olympiad Algebra IV: Symmetric Functions & the Pointwise Trap › Functional equations: the pointwise trap
The unit-circle setup: Put the circumcircle at the unit circle, so vertices $a, b, c$ satisfy $\bar a = 1/a$ .; Olympiad Geometry IV: Barycentric & Complex Coordinates › Complex Numbers in Geometry
The Vieta jump: To prove something about a symmetric equation that is quadratic in each variable (such as $a^2+b^2=k(ab+1)$ ), suppose it fails and take a solution in positive integers with $a+b$ minimal and $a\ge b$ .; Olympiad Number Theory › Vieta Jumping in Full
The write-up checklist: State what you will prove.; USAMO Practice Tests & Write-Up Playbook › Sit a Practice Test
Three standard openings: For a finite point set in the plane: (1) cut the region into pieces so small that two points in one piece do what you want, and count pieces against points; (2) take the convex hull and split into cases by how many points it has; (3) choose an extremal pair…; Olympiad Combinatorics IV: Graphs, Double Counting, Games & Combinatorial Geometry › Points in the Plane: Regions, Hulls, Extremes
Time budget for a 4.5-hour session: Read + small cases on all three: 30 min.; Proof Mastery: Exposition, Lemmas & Scratchpad Strategy › Scratchpad Strategy
Titu’s lemma (Cauchy, Engel form): $\dfrac{x_1^2}{a_1}+\dfrac{x_2^2}{a_2}+\cdots+\dfrac{x_n^2}{a_n}\ \ge\ \dfrac{(x_1+x_2+\cdots+x_n)^2}{a_1+a_2+\cdots+a_n}$ for positive denominators, with equality when the $\tfrac{x_i}{a_i}$ are all equal.; Inequalities › Good-to-Know Inequalities
Trees, components and connectivity: A graph on $n$ vertices with $c$ components and no cycle has exactly $n - c$ edges.; Olympiad Combinatorics V: Graphs, Set Systems & the Probabilistic Method › Graphs: connectivity, extremal edges, planarity
Trig Ceva: concurrency without lengths: Cevians $AD$ , $BE$ , $CF$ of triangle $ABC$ concur iff $\dfrac{\sin\angle BAD}{\sin\angle DAC}\cdot\dfrac{\sin\angle CBE}{\sin\angle EBA}\cdot\dfrac{\sin\angle ACF}{\sin\angle FCB} = 1$ .; Olympiad Geometry V: Directed Angles, Projective Tools & Named Configurations › Directed angles and the bash menu
Two factoring principles: Olympiad Number Theory › Factoring: Divisor Pairs and Coprime Pieces
Two questions, two tools: Can the target be reached?; Invariants & Monovariants › Invariants and Monovariants Worked in Full
Vieta jumping: For a symmetric quadratic Diophantine equation, take the solution with $a + b$ minimal; Vieta gives a second root $a' = kb - a$ that yields a smaller solution unless a boundary case holds — which then proves the required structure.; Olympiad Number Theory › The Proof-Level Toolkit
Weighted AM–GM: For positive $x_1,\dots,x_n$ and positive weights $w_1,\dots,w_n$ with $w_1+\cdots+w_n=1$ , \[w_1x_1+w_2x_2+…+w_nx_n ≥ x_1^w_1x_2^w_2… x_n^w_n,\] with equality exactly when $x_1=x_2=\cdots=x_n$ .; Inequalities › Weighted AM–GM and the Equality Case
Well-ordering / minimal counterexample: Every nonempty set of positive integers has a least element, so assuming a smallest counterexample and building a smaller one is a contradiction.; Proof Techniques › The Main Forms
What a 7 contains: A complete solution states the answer, proves that it works, proves that nothing else works, and justifies every step with a reason a grader can check.; USAMO Practice Tests & Write-Up Playbook › One Problem, Two Write-ups
What a first problem looks like in each area: A USAMO problem 1 or 4 is short to state and has a solution of one or two ideas, each of which must be proved.; Everything USAMO Tests › Four Pillars, Four Problems
What injectivity and surjectivity buy: Functional Equations With Tricky Conditions › Injectivity and Surjectivity as Tools
What you need to know: Four pillars, each appearing on every USAMO/USAJMO: (1) ALGEBRA — inequalities (AM–GM, Cauchy–Schwarz, Schur, rearrangement, Jensen/convexity, smoothing, SOS), polynomials (Vieta, roots of unity, integer-coefficient divisibility), functional equations (subs…; Everything USAMO Tests › The Syllabus, Honestly
When to bash: Coordinates, complex numbers, or barycentrics are legitimate when the configuration is computationally tame (a triangle with one circle, fixed ratios).; Olympiad Geometry › The Configurations That Recur
Which primes can divide $x^2+c$ ?: If a prime $q\nmid c$ divides $x^2+c$ , then $-c$ is a quadratic residue mod $q$ .; Olympiad Number Theory IV: Primitive Roots, Residues & Bounding › Quadratic Residues in Diophantine Equations