Olympiad Algebra | Inequalities, Functional Equations and Polynomials

Olympiad algebra: search for structure before expanding

Symmetry tells you what should stay unchanged

If an expression is unchanged when variables are swapped, preserve that symmetry as long as possible. Symmetric sums, products and substitutions can reduce the number of independent quantities. Blind expansion often destroys visible structure and creates algebra without direction.

Factorisation is a proof tool

A factorised expression exposes zeros, signs and divisibility. Difference of squares, grouping, polynomial identities and substitutions can turn an opaque equation into cases. Before expanding a product, ask whether the factors themselves carry the information the problem needs.

Inequalities begin with domain and equality

Check whether variables are positive, non-negative, real or constrained by a fixed sum/product. Then ask where equality is expected. A target equality at a=b may suggest rewriting around (a-b)². Named inequalities such as AM-GM are useful only when their conditions and equality cases fit.

Normalisation removes irrelevant scale

If a problem is homogeneous, scaling all variables may not change the essential relationship. Setting a sum to 1 or fixing one variable can simplify the search. The proof must justify why the normalisation loses no cases.

Functional equations are systems of constraints

Substitute special values such as 0, 1, -1 or equal variables; swap variables; compare resulting equations; test injectivity or surjectivity only when supported. The objective is to force values or structural properties of the unknown function. Guessing a likely function is useful for discovery but verification and uniqueness still need proof.

Polynomials connect roots, coefficients and factor structure

If P(a)=0, then x-a is a factor. Vieta’s relationships connect sums and products of roots to coefficients. Remainder arguments can evaluate divisibility without full expansion. Olympiad problems often hide an integer or symmetry condition inside polynomial language.

Substitution should reduce complexity

A good substitution groups repeated structure, enforces a constraint or turns an expression into a known form. Introducing a new variable that merely renames an old expression without simplifying relationships adds notation but not insight.

Worked structural example

For positive x and y with x+y fixed, xy is maximised when x=y. One route uses (x-y)²≥0, giving (x+y)²≥4xy. The equality condition is x=y. The important lesson is not the formula but the workflow: identify the fixed symmetric quantity, predict the equality case and choose a representation that makes non-negativity visible.

Common failures

Typical errors include squaring both sides without checking signs, dividing by an expression that may be zero, applying an inequality outside its domain, finding candidate functional-equation solutions without proving completeness, and expanding until the original symmetry disappears.

Transfer test

A student owns olympiad algebra when an unfamiliar expression triggers questions about symmetry, factors, equality, homogeneity, useful substitutions, root structure and domain restrictions before long calculation begins.