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Expansion, factoring, and the standard algebraic identities. Difference of squares: a² − b² = (a − b)(a + b). Sum and difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²). The binomial expansion (a + b)ⁿ via Pascal’s triangle and binomial coefficients. Symmetric expressions and when to expand versus factor.

The discriminant D = b² − 4ac as the central object of quadratic analysis. Three cases: D > 0 gives two distinct real roots, D = 0 gives one repeated root, D < 0 gives no real roots (two complex roots at the AMC 10 level). Computing roots via x = (−b ± √D)/2a once D is determined. Vieta’s formulas as the natural companion: for ax² + bx + c = 0, the sum of roots is −b/a and the product is c/a. Polynomial division and the remainder theorem. Introduction to cubic and higher-degree polynomial behavior at the AMC 10 level.

Setting up systems of equations from word problems. Speed-distance-time, work-rate, mixture, and age problems. The Russian Math “structure first” approach to translating descriptions into equations.

Arithmetic sequences with sum S_n = n(a₁ + a_n)/2. Geometric sequences with sum S_n = a₁(1 − rⁿ)/(1 − r). Telescoping sums and recognizing collapsing structures. Recursive sequences and pattern recognition. Fibonacci-style problems.

Function notation, domain and range. Linear and quadratic functions. Composition of functions: (f ∘ g)(x) = f(g(x)). Inverse functions. Reading information from graphs.

Properties of exponents and logarithms: log(ab) = log(a) + log(b), log(aⁿ) = n · log(a), and change of base log_b(x) = log(x)/log(b). Solving exponential and logarithmic equations. Applications to growth and decay problems.

Linear and quadratic inequalities. Absolute value inequalities. The AM-GM inequality at the AMC 10 level: for non-negative reals, (a + b)/2 ≥ √(ab), with equality when a = b. Recognizing when inequality techniques apply.

Divisibility rules. Prime factorization. Number of factors formula: if n = p₁^a₁ · p₂^a₂ · … · p_k^a_k, then n has (a₁ + 1)(a₂ + 1)…(a_k + 1) divisors. Sum and product of factors formulas. Counting divisors with specific properties.

GCD and LCM via prime factorization. The relationship gcd(a, b) · lcm(a, b) = a · b. The Euclidean algorithm. Setting up and solving simple linear Diophantine equations.

Modular arithmetic mechanics. Solving congruences. Remainder problems. Last-digit and last-two-digit problems via mod 10 and mod 100. Cyclic patterns in modular arithmetic.

Fermat’s Little Theorem: if p is prime and gcd(a, p) = 1, then a^p−1 ≡ 1 (mod p). Introduction to the Chinese Remainder Theorem. Orders and cycles in modular systems.

Place value problems. Number base conversions. Digit-sum problems. Patterns in digit manipulation.

Angle chasing fundamentals: vertical, supplementary, complementary, and corresponding angles. Triangle angle sum (180°) and exterior angle theorem. Properties of isosceles, equilateral, and right triangles.

SSS, SAS, ASA, and AAS congruence. AA, SAS, and SSS similarity. Setting up similar triangles to solve length and area problems. Ratios of corresponding sides and corresponding areas (area scales as the square of the linear ratio).

The Pythagorean theorem: a² + b² = c² for legs a, b and hypotenuse c. Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25). The 30-60-90 triangle (sides in ratio 1 : √3 : 2) and the 45-45-90 triangle (sides in ratio 1 : 1 : √2). Solving for unknown lengths in compound figures.

Properties of parallelograms, rhombuses, rectangles, squares, and trapezoids. Cyclic quadrilaterals and Ptolemy’s theorem at the AMC 10 level. Sum of interior angles of an n-gon: (n − 2) · 180°. Sum of exterior angles always 360°. Regular polygon geometry.

Properties of chords, tangents, and secants. The inscribed angle theorem: an inscribed angle is half the central angle subtending the same arc. Power of a Point: for a point P and circle, if a line through P intersects the circle at A and B, then PA · PB is constant for all such lines. Tangent-chord angle relationships.

Area of a triangle: multiple formulas including (1/2)bh, (1/2)ab · sin(C), abc/(4R) where R is the circumradius, and rs where r is the inradius and s is the semi-perimeter. Heron’s formula: area = √(s(s − a)(s − b)(s − c)) where s = (a + b + c)/2. Shaded region problems. Volumes and surface areas of common 3D figures.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2). Equations of lines and circles: (x − h)² + (y − k)² = r² for a circle centered at (h, k) with radius r. Slope and parallel/perpendicular relationships. Reflections, translations, and rotations in the coordinate plane.

Right triangle trigonometry: sin, cos, tan ratios. Values at common angles (30°, 45°, 60°, 90°). The Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R. The Law of Cosines: c² = a² + b² − 2ab · cos(C). Applications to non-right triangles.

Volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. Volume of a sphere: (4/3)πr³. Volume of a cone: (1/3)πr²h. Cross-sections of 3D figures. Spatial reasoning problems.

The fundamental counting principle. Permutations: P(n, k) = n!/(n − k)!. Combinations: C(n, k) = n!/(k!(n − k)!). Factorials, binomial coefficients, and Pascal’s triangle. Choosing the right counting framework for a problem.

Complementary counting (count what you don’t want, subtract from the total). Casework and systematic enumeration. Counting with restrictions. Recognizing when a problem calls for inclusion-exclusion.

Stars and bars: the number of ways to put n identical objects into k distinct bins is C(n + k − 1, k − 1). The Principle of Inclusion-Exclusion: |A ∪ B| = |A| + |B| − |A ∩ B|, with extensions to three or more sets. Counting arrangements with adjacency or separation constraints.

Setting up recursive counting arguments. Recognizing recursive structures. Solving simple recurrence relations. Classic problems like counting tilings, paths, and arrangements where the answer at step n depends on smaller steps.

Basic probability as favorable outcomes over total outcomes. Probability with combinations. Conditional probability: P(A | B) = P(A ∩ B)/P(B). Geometric probability and continuous probability problems.

Definition: E[X] = Σ x · P(X = x). Linearity of expectation: E[X + Y] = E[X] + E[Y], even when X and Y are dependent. Calculating expected values for simple discrete random variables. Introduction to expected-value reasoning on competition problems.