Formulas & Methods

Study Sheet

Formulas & Methods

The AIME toolkit: every formula, and the method that makes it a solution

Number Theory & Algebra

Tip
Number theory

CRT; orders and primitive roots; φ\varphi, τ\tau, σ\sigma multiplicative; Legendre and Kummer; LTE vp(an−bn)=vp(a−b)+vp(n)v_p(a^n - b^n) = v_p(a - b) + v_p(n); Wilson; quadratic residues (−1-1 QR iff p≡1 mod 4p \equiv 1 \bmod 4); base-bb digit sums ≡n(modb−1)\equiv n \pmod{b-1}; gcd⁡(am−1,an−1)=agcd⁡(m,n)−1\gcd(a^m - 1, a^n - 1) = a^{\gcd(m,n)} - 1; SFFT (x+a)(y+b)=c+ab(x + a)(y + b) = c + ab.

For "remainder when divided by 10001000": CRT with 88 and 125125. For "how many n≤Nn \le N": count by residue classes or by inclusion–exclusion. For Diophantine equations: factor (SFFT), bound, reduce mod a small prime, or descend.

Tip
Algebra and polynomials

Newton's sums; roots of unity filter; ∏k=1n−1(1−ωk)=n\prod_{k=1}^{n-1}(1 - \omega^k) = n and ∏sin⁡kπn=n2n−1\prod\sin\tfrac{k\pi}{n} = \tfrac{n}{2^{n-1}}; Lagrange interpolation; finite differences (degree-dd polynomial has constant ddth differences); floor: ∑⌊k/m⌋\sum\lfloor k/m\rfloor by blocks, ⌊n⌋\lfloor\sqrt n\rfloor constant on square blocks; {x}\{x\} equations by casework on ⌊x⌋\lfloor x\rfloor; symmetric substitutions u=x+1xu = x + \tfrac1x; telescoping and periodic recurrences.

For values of a polynomial at consecutive integers, use finite differences or an auxiliary polynomial with known roots. For floor equations, set n=⌊x⌋n = \lfloor x\rfloor and solve n≤x<n+1n \le x < n+1 per case.

Counting, Probability & Expectation

Tip
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 1∣G∣∑∣Fix(g)∣\tfrac{1}{|G|}\sum|\mathrm{Fix}(g)|; Vandermonde and hockey stick; distributing with restrictions via complements.

State the recurrence, compute small cases, and identify (Fibonacci, Catalan, tribonacci) before closing the form. Bounded stars-and-bars: subtract the overshoot cases.

Tip
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 for "equally likely orderings."

Define the states and write Ei=1+∑pijEjE_i = 1 + \sum p_{ij}E_j. For continuous problems draw the region and compute areas; for "same color/pair" problems count favorable pairs directly.

Geometry & Trigonometry

Tip
Geometry

Power of a point; radical axis; Ptolemy; Brahmagupta; Stewart; mass points and Ceva/Menelaus; Heron; rs=abc4Rrs = \tfrac{abc}{4R}; Euler OI2=R(R−2r)OI^2 = R(R - 2r); area ratios via shared angles [ADE][ABC]=AD⋅AEAB⋅AC\tfrac{[ADE]}{[ABC]} = \tfrac{AD\cdot AE}{AB\cdot AC}; coordinates with a smart origin; complex numbers on the unit circle for cyclic configurations; 3D: distances via coordinates, volumes by 13Bh\tfrac13Bh, cross-sections.

Compute the area two or three ways. Put the circumcenter at the origin when a circle dominates. For ratio-chasing, mass points first, coordinates second.

Tip
Trigonometry

Product-to-sum and sum-to-product; tan⁡\tan addition; ∑k=1n−1cos⁡2kπn=−1\sum_{k=1}^{n-1}\cos\tfrac{2k\pi}{n} = -1; cos⁡π7cos⁡2π7cos⁡3π7=18\cos\tfrac\pi7\cos\tfrac{2\pi}7\cos\tfrac{3\pi}7 = \tfrac18 (doubling trick); telescoping via tan⁡(a)−tan⁡(b)=sin⁡(a−b)cos⁡acos⁡b\tan(a) - \tan(b) = \tfrac{\sin(a - b)}{\cos a\cos b}; sin⁡xsin⁡(60−x)sin⁡(60+x)=14sin⁡3x\sin x\sin(60 - x)\sin(60 + x) = \tfrac14\sin 3x; De Moivre for cos⁡nθ\cos n\theta polynomials.

Turn products into sums, sums of cosines into roots of unity, and telescope whenever consecutive angles appear. Multiply by sin⁡\sin of the smallest angle to trigger the doubling trick.

Concept
The universal problem-solving loop

Read twice. Restate the goal in your own words. List the givens. Pick a representation. Try small cases. Look for symmetry, an invariant, or an extremal object. Compute, then CHECK against a second method or a sanity bound.

At AMC 10/12 and AIME level, the second method is what separates a 44-second guess from a 1212-second certainty: compute a probability two ways (complement and direct), a length via two theorems, a count via a recurrence and a formula. Casework must be organized by a stated criterion so nothing is double-counted; algebra must be checked by substituting back.