Pre-Olympiad Competition Math – Key Formulas & Theorems

58 vocabulary-style flashcards summarizing essential geometry, number theory, algebra, and counting/probability formulas for pre-Olympiad competition math. Each card pairs a concise term with its defining formula or theorem for quick review.

Area of a Triangle

A = ½bh = rs = ½ab sin θ = abc⁄4R (multiple equivalent forms).

Area of a Square

A = s² or, for perpendicular diagonals, A = d₁d₂⁄2.

Area of a Rectangle

A = bh (base × height).

Area of a Trapezoid

A = (b₁ + b₂)h⁄2.

Area of a Regular Hexagon

A = 3√3 s²⁄2.

Area of a Regular Polygon

A = ap⁄2 = n s²⁄[4 tan(180°/n)].

Volume of a Cone

V = (πr²h)⁄3; Surface Area = πr² + πrl.

Volume of a Sphere

V = 4πr³⁄3; Surface Area = 4πr².

Volume of a Cube

V = s³; Surface Area = 6s².

Volume of a Pyramid

V = ⅓bh; Surface Area = 2sl + b (square base).

Volume of a Cylinder

V = πr²h; Surface Area = 2πr² + 2πrh.

Volume of a Prism

V = lwh; Surface Area = 2(lw + lh + wh).

Pythagorean Theorem

a² + b² = c²; includes special 45-45-90 and 30-60-90 ratios and common triples.

Distance Formula

d = √[(x₂ – x₁)² + (y₂ – y₁)²].

Heron’s Formula

Area Δ = √[s(s – a)(s – b)(s – c)].

Cyclic Quadrilateral Properties

Opposite angles sum to 180°; perpendicular bisectors concur at circumcenter; equal angle subtensions.

Ptolemy’s Theorem

For cyclic ABCD: ac + bd = ef (sides a,c; b,d; diagonals e,f).

Brahmagupta’s Formula

Area of cyclic quad: K = √[(s – a)(s – b)(s – c)(s – d)].

Power of a Point

Products of chord/tangent segments are equal (AE·EC = DE·EB, etc.).

Ceva’s Theorem

Lines AD, BE, CF concur iff (BD/DC)(CE/EA)(AF/FB) = 1.

Menelaus’ Theorem

For transversal PQR on ΔABC: (BP/PC)(CQ/QA)(AR/RB) = 1.

Circle Arcs & Angles

Central angle = intercepted arc; inscribed angle = ½ intercepted arc.

Angle Bisector Theorem

In ΔABC, AD bisects ∠A ⇒ BD/DC = AB/AC.

Basic Trig Identities

Includes tan = sin/cos, double-angle, Pythagorean, addition–subtraction, half-angle formulas.

Triangle Inequality

Sum of any two sides of a triangle exceeds the third.

Pick’s Theorem

Area = I + ½B – 1 for lattice polygons.

Stewart’s Theorem

In ΔABC with cevian AD: man + dad = bmb + cnc (BD = m, DC = n, AD = d).

Extended Law of Sines

a⁄sin A = b⁄sin B = c⁄sin C = 2R (circumradius R).

Law of Cosines

c² = a² + b² – 2ab cos C (and cyclic permutations).

Shoelace Theorem

Polygon area = ½ |Σxᵢyᵢ₊₁ – Σyᵢxᵢ₊₁| for ordered vertices.

Sum of an Arithmetic Series

Sₙ = n⁄2 (a₁ + aₙ).

Finite Geometric Series Sum

S = a₁(1 – rⁿ)/(1 – r), |r| < 1.

Infinite Geometric Series Sum

S = a₁/(1 – r), |r| < 1.

Sums of First n Integers

Odd = n²; Even = n(n + 1); All = n(n + 1)/2.

Divisor Count & Sum

For n = p₁ˣ p₂ʸ …, #divisors = (x + 1)(y + 1)…; sum = Π(1 + p + … + pˣ).

Chinese Remainder Theorem

System of congruences with coprime moduli has unique solution mod product.

Chicken McNugget Theorem

Largest unattainable ax + by (gcd=1) is ab – a – b; count is (a – 1)(b – 1)/2.

Euler’s Totient Function

φ(n) = n Π(1 – 1/p) over distinct primes p|n; gives count of units mod n.

Wilson’s Theorem

For prime p, (p – 1)! ≡ –1 (mod p).

Trivial Inequality

x² ≥ 0 for all real x (foundation of many inequalities).

Fibonacci Numbers

F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂; ratios approach (1 + √5)/2.

Pigeonhole Principle

n items in k boxes with n > k ⇒ some box contains ≥2 items.

Logarithm Rules

Conversion, addition, subtraction, power, change-of-base, reciprocal identities for logs.

Vieta’s Formulas

Relate polynomial coefficients to sums/products of roots (e.g., ax² + bx + c: r₁ + r₂ = –b/a).

Common Factorizations

a² – b², a³ ± b³ identities; SFFT technique for integer solutions.

Quadratic Formula

x = [–b ± √(b² – 4ac)]/(2a); discriminant determines root nature.

RMS ≥ AM ≥ GM ≥ HM

Inequality chain of means; equality when all numbers equal.

DeMoivre’s Theorem

[r cis θ]ⁿ = rⁿ cis (nθ); useful for complex powers/roots.

Permutations

nPₖ = n!/(n – k)! (ordered selections).

Combinations

C(n,k) = n!/[k!(n – k)!] (unordered selections).

Pascal’s Identity

C(n,k) = C(n – 1,k – 1) + C(n – 1,k).

Hockey-Stick Identity

Σ_{i=r}^{n} C(i,r) = C(n + 1, r + 1).

Binomial Theorem

(a + b)ⁿ = Σ_{k=0}^{n} C(n,k) a^{n–k} b^{k}; row sum 2ⁿ.

Fundamental Counting Principle

Total outcomes = product of counts of sequential choices.

Burnside’s Lemma

Total distinct arrangements = (1/|G|) Σ |Fix(g)| over group actions g.

Stars and Bars

Distribute n identical items into k boxes: C(n – 1, k – 1) (non-empty) or C(n + k – 1, k – 1) (allow empty).

Expected Value

E(X) = Σ P(Xᵢ)·V(Xᵢ); linearity: E(X + Y) = E(X) + E(Y).