Competition Math

The contest toolbox: Vieta's formulas, telescoping, invariants, the extremal principle, coloring arguments, and functional equations.

Vieta's Formulas — r + s = -b, rs = c — work with roots without ever finding them.
Telescoping Sums — Write each term as a_k - a_k+1 and watch the whole sum collapse.
Invariants & Monovariants — Find what a move can never change — and rule out entire futures at once.
The Extremal Principle — Interrogate the largest, smallest, or closest object — extremes can't be improved.
Coloring Arguments — Color the board so the tiles can't help but disagree with the totals.
Functional Equations — Plug in the right values — x = 0, y = -x, x 1/x — and the function confesses.

Further Topics

Nim and Sprague-Grundy Values — XOR the pile sizes to win Nim; mex turns any impartial game into one.
Infinite Descent — Build a smaller counterexample; the positive integers can't descend forever.
Double Counting — Count one set two ways and equate the totals.
Roots of Unity Filter — Average a polynomial over roots of unity to sum coefficients in one residue class.
Chicken McNugget (Frobenius) Theorem — Largest unreachable total for two coprime coins: ab - a - b.
Substitution in Inequalities — Normalize, Ravi, and trig substitution to reach AM-GM form.
Wythoff's Game — Two piles, one rule, and losing positions ruled by the golden ratio.