What the AMC 10 Tests
Everything on the AMC 8, plus: quadratics and Vieta; polynomial remainder/factor theorems; systems of equations; exponents and radicals; basic logarithms (rarely) and no trigonometry; modular arithmetic, bases, divisor counting; counting with inclusion–exclusion, stars and bars, casework and complements; probability with conditional reasoning and expected value basics; geometry with similar triangles, power of a point, inscribed angles, area ratios, coordinate and 3D geometry (volumes, surface areas); sequences (arithmetic, geometric, telescoping). Twenty-five questions, seventy-five minutes.
Gap Fillers
Cube diagonals (), cone and pyramid volumes ( base height), sphere (, surface ), and "unfold the surface" for shortest paths on boxes. A cube with surface area has edge and volume .
Linearity: with NO independence needed. The expected number of fixed points of a random permutation is ; the expected sum of two dice is ; "expected number of steps" problems set up first-step equations .
Inscribed angle is half its arc; angles from the same arc are equal; a tangent makes a right angle with the radius; the angle between a tangent and a chord equals the inscribed angle on the other side. Power of a point () converts lengths on secants and tangents.