Algebra for Math Contests (AMC, AIME, IMO)

Explore the nonnegativity of real number squares, apply sum of squares techniques to Olympiad problems, and learn problem-solving methods for equations and systems using this key inequality.

Explore ten inequality problems from the olympiad algebra course. Apply binomial theorem, completing the square, and sum-of-squares techniques to prove bounds and equality cases.

Explore inequality techniques using sum of squares and perfect-square rewrites to solve multi-variable problems and contest-style real-number challenges.

Explore olympiad algebra inequalities, including am-gm-hm-qm inequalities, rearrangement, chebyshev, nesbitt, cauchy–schwarz, titu's lemma, power means, minkowski, and schur, with problem-solving applications.

Apply olympiad inequality techniques to problems on inequalities, using the arithmetic mean–geometric mean inequality, harmonic–arithmetic mean relation, Cauchy and Titu's lemma, plus substitutions, to derive bounds and equality cases.

Explore maxima and minima of algebraic expressions with multiple variables using non-calculus techniques, including sum-of-squares, AM-GM, QM-AM, and Cauchy, through Olympiad-style problems.

Continue mastering the extrema of algebraic expressions without calculus, using tricks to reduce to two variables, apply discriminants and Cauchy inequality, and solve olympiad problems.

Explore the cube identity a^3+b^3+c^3−3abc = (a+b+c)(a^2+b^2+c^2−ab−bc−ca) with two proofs and useful variants, and apply it to olympiad problems and AM-GM consequences in contest algebra.

This lecture applies the cubic identity a^3+b^3+c^3-3abc to problems on identities, consecutive integers, and triangle inequalities; it also explores solving diophantine equations, cube-root constructions, and equilateral triangles on algebraic curves.

Apply algebraic identities, including a^3+b^3+c^3-3abc and related factorization. Solve contest problems on roots and integer equations using these techniques.

Explore Lagrange's identity and its complex-number interpretation, showing how products of sums of squares become sums of squares, with Olympiad-style applications.

Explore Sophie Germain's identity for algebraic factorization with a^4+4b^4, apply Lagrange's identity, and solve olympiad-style problems to master divisibility and factorization techniques.

Learn reciprocal polynomials and their properties in olympiad problem solving, including self-reciprocal forms. Apply these techniques to problems with non-negative coefficients, degree considerations, and find the greatest value of p(3)/p(2).

Compute the sum of squares of polynomial coefficients using the coefficient of x^0 in P(x)P(1/x), with P2N and Q2N, and with x to minus one over x for alternating signs.

Explore how nonzero roots invert to become the roots of the reciprocal polynomial, using the reciprocal polynomial theorem and practice with contest-style problems and limits.

Explore self-reciprocal polynomials, define the condition p(x)=x^d p(1/x), and examine even and odd degree cases, coefficient symmetry, and root-pairing properties with examples.

Explore how polynomials with real coefficients interact with conjugates, demonstrating p(z) conjugate equals p(z conjugate) and |p(z)|^2 equals p(z) p(z conjugate) via cube roots of unity.

Explore roots of unity and complex polynomials using de moivre and unit-circle angles to bound coefficient sums via g of theta.

Explore roots of polynomials with real coefficients, show complex roots come in conjugate pairs, and use quadratic and reciprocal polynomials to deduce divisibility and root modulus properties.

Explore six contest-style problems on roots of polynomials and complex numbers. Learn techniques including factorization, Vieta's formulas, root conjugates, roots of unity, and Jensen's inequality.

Explore olympiad polynomial problems: prove all nonzero roots of a degree-four p(x) lie on unit circle, and find a such that p(x) is divisible by (x−a)^2, giving a = ±√2.

Explore triangle inequality and complex number techniques in polynomial problems, applying de Moivre's theorem and trigonometry to unit-circle roots and modulus bounds.

Use the triangle inequality to bound polynomial roots by isolating a term and taking moduli. Apply these bounds to contest problems and the unit-circle root lemma for positive-coefficient polynomials.