Olympiad Algebra & Functional Equations

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 squares) method. Always locate the equality case first; it tells you which inequality to apply and where to apply it.

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 $\tfrac{x^2}{2} + cx$). Cauchy's equation $f(x+y) = f(x) + f(y)$ has only linear solutions under any regularity (monotone, bounded, continuous). Always verify the found function satisfies the original equation.

Integer polynomials: $a - b \mid P(a) - P(b)$. Roots of unity filter and symmetric-function bookkeeping ($r^2 + s^2 + t^2 = 4$ for $x^3 - 2x + 5$ without touching the roots). Irreducibility via Eisenstein or reduction mod $p$. Degree and leading-coefficient comparison to pin down polynomial identities.