ALGEBRA
Equations, inequalities, polynomials, functions, and sequences — the largest single category on AMC10.
What Algebra means on the AMC10
Roughly a third of every AMC10 paper tests algebraic reasoning. You will see:
- Linear and quadratic equations and systems of them
- Polynomials — factoring, roots, Vieta's formulas
- Exponents and logarithms — rules and clever rewrites
- Functions — domain/range, composition, simple transformations
- Sequences and series — arithmetic, geometric, telescoping
The key skill is noticing structure so you can pick the fast solution. Many AMC10 algebra problems have an "ugly" path and a one-line path; the goal of this section is to train your eye for the second.
How to use this section
Pick a subtopic, read the concept page, then jump into Practice. Each problem includes the primary solution and (where useful) an alternate method.
SUBTOPICS
Mastering exponent laws, fractional and negative exponents, and the definition and basic laws of logarithms for the AMC10.
Understanding function notation, composition, inverses, and graph transformations for AMC10 success.
Techniques for solving equations and inequalities that involve variables to the first power, including systems of two equations and absolute-value conditions.
Master polynomial arithmetic, the Factor and Remainder Theorems, Vieta's formulas for cubics, and evaluation tricks used throughout AMC 10.
Solving and reasoning about quadratic equations on the AMC10.
Arithmetic and geometric sequences, series formulas, telescoping sums, and recursive patterns at AMC10 level.
Translating real-world scenarios into equations and solving them with systematic algebra at AMC10 level.