Olympiad Algebra

Advanced → Olympiad

Competition algebra is not a bag of tricks. It is a handful of ideas that keep coming back in new disguises. Each chapter builds one idea on a figure, then sets it loose on contest problems.

Assumes fluent school algebra. Start with Algebra Studio →

1
Symmetry
Sum and product
Anything that doesn't change when a and b swap places can be written with a + b and ab alone. You never need a or b.
Case: The seventh power
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2
Inequalities
Never negative
A square is never below zero. From that one fact: completing the square, AM–GM, and why equal parts win.
Case: The biggest box
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3
Vieta
Roots without solving
A polynomial's coefficients already know the sum and product of its roots.
Case: The roots you never meet
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4
Homogenization
Same degree everywhere
When every term has the same degree, scale doesn't matter: fix one variable, or use the condition to lift every term to the same degree.
Case: One condition, any scale
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5
Polynomials
Pinned down by very little
Remainders, values and roots: a polynomial is fixed by far fewer facts than it seems.
Case: Guess the polynomial
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6
Recurrences
Sequences that feed themselves
Each term from the ones before: fixed points, closed forms and cycles.
Case: The sequence that comes home
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7
Functional equations
Functions from clues
Feed a function clever inputs until it gives itself away.
Case: Triangular by force
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8
Invariants
What never changes
When a process scrambles numbers, look for the quantity it can't touch.
Case: The blackboard game
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Final round

Ten problems from the whole studio. Nothing tells you which idea to use.