Search & glossary

Every rule and concept in AMC 10, with its definition and a link to the lesson that teaches it. Search by term, by idea, or by lesson title.

Glossary (64)

3D geometry: Cube diagonals ( $s\sqrt3$ ), cone and pyramid volumes ( $\tfrac13$ base $\times$ height), sphere ( $\tfrac43\pi r^3$ , surface $4\pi r^2$ ), and "unfold the surface" for shortest paths on boxes.; Complete Syllabus & Gap Fillers › Gap Fillers
AM–GM and when equality holds: For nonnegative $a_i$ : $\dfrac{a_1+\cdots+a_n}{n} \ge \sqrt[n]{a_1\cdots a_n}$ , with equality exactly when all $a_i$ are equal.; Algebraic Weapons › Inequalities That Decide Problems
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 series: proof of $S_n = \tfrac{n}{2}(a_1 + a_n)$: Write the series forwards and backwards and add: each column gives $a_1 + a_n$ (what one end gains, the other loses), so $2S_n = n(a_1 + a_n)$ .; Where the Formulas Come From › The Series Formulas, Derived
Arrangements with repeated letters: $\dfrac{n!}{n_1!\,n_2!\cdots}$ for a word with repeated letters; divide by the arrangements of the identical items.; Counting: Inclusion–Exclusion & Stars and Bars › Distributions, Arrangements and Overcounting
Cauchy–Schwarz in the useful form: $(a_1^2+\cdots+a_n^2)(b_1^2+\cdots+b_n^2) \ge (a_1b_1+\cdots+a_nb_n)^2$ .; Algebraic Weapons › Inequalities That Decide Problems
Circles and polygons: Inscribed angle $= \tfrac12$ arc; tangent $\perp$ radius; power of a point $PA\cdot PB = PC\cdot PD = PT^2$ ; chord length $2\sqrt{r^2 - d^2}$ .; Formulas & Methods › Geometry
Circles: the four angle facts: Inscribed angle is half its arc; angles from the same arc are equal; a tangent makes a right angle with the radius; the angle between a tangent and a chord equals the inscribed angle on the other side.; Complete Syllabus & Gap Fillers › Gap Fillers
Congruence arithmetic: $a\equiv b\pmod n$ means $n\mid(a-b)$ .; Number Theory: Modular Arithmetic › Working Modulo n
Coordinates and 3D: Distance, midpoint, slope; perpendicular slopes multiply to $-1$ ; point-to-line $\tfrac{|ax_0 + by_0 + c|}{\sqrt{a^2+b^2}}$ ; shoelace for polygon area.; Formulas & Methods › Geometry
Counting: Permutations, combinations, $\binom nk = \binom{n}{n-k}$ , Pascal $\binom nk = \binom{n-1}{k-1} + \binom{n-1}{k}$ .; Formulas & Methods › Counting, Probability & Number Theory
Denest $\sqrt{m + 2\sqrt n}$ and clear radicals with conjugates: If $x + y = m$ and $xy = n$ with $x \ge y \ge 0$ , then $\sqrt{m + 2\sqrt n} = \sqrt x + \sqrt y$ and $\sqrt{m - 2\sqrt n} = \sqrt x - \sqrt y$ .; Algebraic Weapons › Radicals: Conjugates and Denesting
Derivation by completing the square: From $ax^2 + bx + c = 0$ : divide by $a$ , move the constant, and complete the square: $\left(x + \dfrac{b}{2a}\right)^2 = \dfrac{b^2 - 4ac}{4a^2}$ .; Where the Formulas Come From › The Quadratic Formula, Derived
Derivation: the number of subsets is $2^n$: A set with $n$ elements has $2^n$ subsets, and $\sum_{k}\dbinom nk = 2^n$ .; Where the Formulas Come From › Two More Derivations Worth Knowing
Derivation: the sum of the first $n$ squares: $1^2 + 2^2 + \cdots + n^2 = \dfrac{n(n+1)(2n+1)}{6}$ .; Where the Formulas Come From › Two More Derivations Worth Knowing
Difference of squares, everywhere: $a^2 - b^2 = (a - b)(a + b)$ — the most-used identity in competition algebra.; Algebraic Weapons › Identities That Do the Work
Distributing identical objects: Nonnegative solutions of $x_1+\cdots+x_k=n$ number $\dbinom{n+k-1}{k-1}$ ; positive solutions number $\dbinom{n-1}{k-1}$ .; Counting: Inclusion–Exclusion & Stars and Bars › Stars and Bars
Expected value, the fast way: Linearity: $E[X + Y] = E[X] + E[Y]$ with NO independence needed.; Complete Syllabus & Gap Fillers › Gap Fillers
Exponents, radicals, logs (light): $a^ma^n = a^{m+n}$ , $(a^m)^n = a^{mn}$ , $a^{-n} = \tfrac{1}{a^n}$ , $a^{1/n} = \sqrt[n]a$ .; Formulas & Methods › Algebra
Fermat and Euler for exponents: $a^{p-1} \equiv 1 \pmod p$ for a prime $p \nmid a$ ; more generally $a^{\varphi(n)} \equiv 1 \pmod n$ when $\gcd(a,n) = 1$ , with $\varphi(100) = 40$ .; Number Theory: Modular Arithmetic › Chinese Remainder and Last-Two-Digit Problems
Fermat's little theorem: If $p$ is prime and $p\nmid a$ , then $a^{p-1}\equiv 1\pmod p$ .; Number Theory: Modular Arithmetic › Working Modulo n
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 series: proof of $S_n = a\,\dfrac{1 - r^n}{1 - r}$: Multiply the sum by $r$ and subtract: $S_n - rS_n = a - ar^n$ — every middle term appears once in each copy and cancels.; Where the Formulas Come From › The Series Formulas, Derived
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
Manipulate without solving: Symmetric expressions reduce to $S = r+s$ and $P = rs$ : $r^2+s^2 = S^2 - 2P$ and $\tfrac1r+\tfrac1s = \tfrac SP$ .; Algebra & Polynomials › Polynomial Roots
Mass points: Hang weights at the vertices so each side balances at its division point (weight $\times$ length equal on both sides); the weights reveal all segment ratios.; Geometry: Power of a Point & Mass Points › Mass Points and Coordinates
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
Number theory: Modular arithmetic; Fermat $a^{p-1} \equiv 1$ ; Euler $a^{\varphi(n)} \equiv 1$ ; $\varphi(n) = n\prod(1 - \tfrac1p)$ .; Formulas & Methods › Counting, Probability & Number Theory
Probability and expected value: Conditional $P(A\mid B) = \tfrac{P(A\cap B)}{P(B)}$ ; independent events multiply.; Formulas & Methods › Counting, Probability & Number Theory
Proof of Power of a Point (two chords): Chords $AB$ and $CD$ meet at $P$ .; Where the Formulas Come From › Circle Power, Proved
Proof of the inscribed angle theorem: First the case where one chord is a diameter: the triangle from the center is isosceles (two radii), so the central angle, an exterior angle of it, equals the sum of two equal base angles — twice the inscribed angle.; Where the Formulas Come From › Circle Power, Proved
Proof of Vieta's formulas: If $r$ and $s$ are the roots, the quadratic factors as $a(x - r)(x - s) = a\left(x^2 - (r+s)x + rs\right)$ .; Where the Formulas Come From › The Quadratic Formula, Derived
Proof: $\dfrac{a + b}{2} \ge \sqrt{ab}$ for $a, b \ge 0$: Squares are never negative: $\left(\sqrt a - \sqrt b\right)^2 \ge 0$ .; Where the Formulas Come From › AM–GM, Proved
Quadratics and Vieta: $x = \tfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ ; roots sum to $-\tfrac ba$ , multiply to $\tfrac ca$ ; vertex at $x = -\tfrac{b}{2a}$ .; Formulas & Methods › Algebra
Ratios: lengths, areas, volumes: Similar figures with ratio $k$ have areas in ratio $k^2$ and volumes in ratio $k^3$ .; Geometry: Power of a Point & Mass Points › Similar Triangles, Areas and the Shoelace Formula
Remainder and factor theorems: The remainder of $P(x)$ divided by $x-a$ is $P(a)$ ; so $x-a$ is a factor exactly when $P(a) = 0$ .; Algebra & Polynomials › Polynomial Roots
Sequences and telescoping: Arithmetic: $S_n = \tfrac{n(a_1 + a_n)}{2}$ .; Formulas & Methods › Algebra
Simon's Favourite Factoring Trick: $xy + ax + by = c \iff (x+b)(y+a) = c + ab$ : add the missing constant and factor.; Algebra & Polynomials › Vieta, Symmetric Sums and Clever Factoring
Squares have few remainders: Every square is $0$ or $1 \pmod 4$ , and $0$ , $1$ or $4 \pmod 8$ ; mod $3$ it is $0$ or $1$ .; Number Theory: Modular Arithmetic › Squares, Remainders and Impossibility
Symmetric sums via Vieta: By Vieta's formulas, everything symmetric in two roots can be written using $x + y$ and $xy$ : for instance $x^2 + y^2 = (x+y)^2 - 2xy$ .; Algebraic Weapons › Identities That Do the Work
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
The checklist: Everything on the AMC 8, plus: quadratics and Vieta; polynomial remainder/factor theorems; systems of equations; exponents and radicals; basic logarithms (rarely) and no trigonometry; modular arithmetic, bases, divisor counting; counting with inclusion–excl…; Complete Syllabus & Gap Fillers › What the AMC 10 Tests
The constant product: From a point $P$ , along any line through $P$ meeting a circle at $A,B$ , the product $PA\cdot PB$ is constant; for a tangent of length $t$ , $t^2 = PA\cdot PB$ .; Geometry: Power of a Point & Mass Points › Power of a Point
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 principle: $|A\cup B\cup C| = |A|+|B|+|C| - |A\cap B| - |A\cap C| - |B\cap C| + |A\cap B\cap C|$ .; Counting: Inclusion–Exclusion & Stars and Bars › Inclusion–Exclusion
The shoelace formula: For vertices $(x_1,y_1), \dots, (x_n,y_n)$ in order, the area is $\tfrac12\left|\sum (x_iy_{i+1} - x_{i+1}y_i)\right|$ .; Geometry: Power of a Point & Mass Points › Similar Triangles, Areas and the Shoelace Formula
The substitution $u = x + \tfrac{1}{x}$: From $u = x + \tfrac1x$ : $x^2 + \tfrac{1}{x^2} = u^2 - 2$ and $x^3 + \tfrac{1}{x^3} = u^3 - 3u$ .; Algebraic Weapons › Identities That Do the Work
The three coordinate workhorses: Distance $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ ; midpoint $\left(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\right)$ ; a circle $(x-h)^2 + (y-k)^2 = r^2$ .; Circles & Lines › Coordinates: Distance, Midpoint, and the Circle Through Three Points
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
Trade $x$ and $y$ for $s = x + y$ and $p = xy$: If a system is unchanged by swapping $x$ and $y$ , rewrite it in $s = x + y$ and $p = xy$ using $x^2 + y^2 = s^2 - 2p$ and $x^3 + y^3 = s^3 - 3ps$ .; Algebra & Polynomials › Symmetric Systems: Solve for the Sum and Product
Triangles: Heron: $\sqrt{s(s-a)(s-b)(s-c)}$ .; Formulas & Methods › Geometry
Two congruences at once: If $\gcd(m,n) = 1$ , the pair $x \equiv a \pmod m$ , $x \equiv b \pmod n$ has exactly one solution modulo $mn$ .; Number Theory: Modular Arithmetic › Chinese Remainder and Last-Two-Digit Problems
Using the equality case on purpose: AM–GM problems are usually about WHERE equality happens: to optimize, arrange the terms so they can all be equal, then read off the extreme value.; Where the Formulas Come From › AM–GM, Proved
Vieta for any degree: For a monic polynomial with roots $r_1,\dots,r_n$ : $\sum r_i = -a_{n-1}$ , $\sum_{i<j} r_ir_j = a_{n-2}$ , and $\prod r_i = (-1)^n a_0$ .; Algebra & Polynomials › Polynomial Roots
Vieta's relations: For $ax^2+bx+c$ : $r_1+r_2 = -\tfrac ba$ , $r_1r_2 = \tfrac ca$ .; Algebra & Polynomials › Vieta, Symmetric Sums and Clever Factoring
When in doubt, use coordinates: Place the figure on axes; use the distance formula, perpendicular slopes, and the shoelace area formula.; Geometry: Power of a Point & Mass Points › Mass Points and Coordinates
When to place the axes yourself: A configuration with a right angle, a midpoint, or a line of symmetry can be placed on the axes for free: put the right angle at the origin and the legs on the axes.; Circles & Lines › Coordinates: Distance, Midpoint, and the Circle Through Three Points
Which formula, and why: Ordered with repetition: $n^k$ .; Counting: Inclusion–Exclusion & Stars and Bars › Distributions, Arrangements and Overcounting
Why the discriminant decides everything: In the derivation, the only square root taken is $\sqrt{b^2 - 4ac}$ .; Where the Formulas Come From › The Quadratic Formula, Derived
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