Introduction to Problem Solving for Math Contests

Master angle chasing in geometry through problem-based lessons, from similarity and congruency to olympiad applications. Build notes and work through a problem set to excel in math contests.

Explore angle chasing and construction strategies for math contests, including reflections, perpendicular bisectors, isosceles and equilateral triangles, SAS and sine-rule proofs, and cyclic quadrilateral reasoning.

Learn angle chasing with exterior angles and the sum of interior opposite angles, and apply triangle inequality and isosceles constructions to compare triangle sides and angles.

Learn angle chasing techniques in a square to prove that angle NBC is twice angle AB, using trigonometry, geometry, and midpoint constructions.

Explore core geometry theorems for math contests, including the midpoint theorem, angle bisector theorem and centroid theorem, and Shiva's theorem and Menelaus theorem, with proofs and applications.

Explore geometry contest problem solving with area comparisons, similar triangles, and angle chasing, illustrated through China, Romania, and olympiad-type questions and tricks like extending figures.

Explore geometric problem solving with isosceles triangles, angle chasing, and constructions like altitudes to reveal congruent triangles; emphasizes practice, notes, and persistent effort.

Learn circle theorems essential for math contests: central angles, equal chords, chord-arc relations, inscribed angles, intersecting and external chords, alternate segment theorem, power of a point, cyclic quadrilaterals, and Ptolemy.

Explore olympiad geometry through Romania, Britain, Ukraine, and Moscow problems, using angle chasing, isosceles triangles, incenter properties, and midpoint theorem strategies.

Tackles two contest problems through angle chasing in cyclic quadrilaterals and isosceles triangles, using midpoints and parallels to determine key angles.

Explore five contest-style problems across number theory, algebra, combinatorics, and geometry. Apply modular arithmetic, complementary counting, trailing zeros in factorials, telescoping products, and Simon's favorite factoring trick to find solutions.

solves problems 6–10 by applying gcd–lcm product relations, completing the square, parity, and modular arithmetic to find integer pairs, primes, and remainders.

Solve five problems from problem 11 to 15, including a telescoping product limit, a greatest-integer and fractional-part equation, triangle geometry with altitude, median and angle bisector, and an AP–GP puzzle.

Master problem solving for math contests through problems on end zeros in powers, finding prime triples, a fourth-root expression, parallel-line geometry with similar triangles, and solving a base-10 logarithm equation.

Explore problem solving strategies for math contests: bound and solve positive-integer triples in problems 21–25, using factorization, bounds, and inequalities like Cauchy-Schwarz and MGM to derive minima and equality cases.

This lecture covers problems 26–30, using substitution to solve a functional equation, then tackles a prime-pair problem, a discriminant-based quadratic, a cube-sum factoring trick, and a symmetry argument yielding x=0.

Explore olympiad problem solving through problems 31–35, using Pitot's theorem and Brahmagupta's formula on cyclic quadrilaterals, Lagrange's identity in algebra, and angle bisector and cosine rules to derive key results.

Explore contest problem solving with problems 36–40, using sum of squares, conjugates, and systems; review later problems 41–43 as homework.

Explore problem solving strategies for contest math, including pattern recognition to factor cubics, circle geometry with inscribed quadrilaterals, trig and half-angle methods, and applying Cauchy and vector inequalities.

Develop contest problem solving through symmetry and squaring tricks, form a single governing equation from two, and solve for integers, positives, and trigonometric cases in problems 46–50.

Solve contest problems 51–55 by applying factorization of diophantine equations, Lagrange's identity to sums of squares, Cauchy's inequality equality cases for trigonometry, and circle geometry with Pythagoras to compute lengths.

Explore problem solving strategies for math contests, including applying Cauchy inequality and equality case, solving equations via substitution and guessing, and using increasing function and sign changes to locate roots.

Develop problem solving techniques for math contests, including pattern guessing for recurrences and divisibility by 101. Apply complex numbers, cosine rule with 60-degree angles, and digit-logic problems.

Explore problem solving tricks for math contests, including mod 27 arithmetic and divisibility proofs, a summation trick with n and k-n, and median problems via Stewart's theorem.

Master problem solving strategies for math contests by squaring and adding to cancel terms, using discriminants to force integer solutions, applying Cauchy–Schwarz, and shifting variables for insights in problems 72–76.

Explore problem solving for math contests through polynomials with integer coefficients, divisibility and roots analysis, prime divisors, and the range of a rational function, with hands-on problems 77–82.

Solve five contest problems (83–87) by applying cubic factorization, substitutions, and polynomial methods using Vieta's relations.

Explore extremal principle in the two boys and two girls dance problem; prove algebraic inequalities via sum of squares and am-gm; solve a diophantine bound; apply pigeonhole geometry on circles.

explore problem solving for math contests through problems 92-96, employing induction, set closure, trigonometric identities, triangle inequalities, and complementary counting to prove key properties.

Problems 97–100 derive a, b, c relation from a quadratic in b with a positive root, use Sophie Germain identity, and prove h ≥ R + r in triangles.

Explore problem solving for math contests through solutions to RMO 2016 problems 101–105, including geometry with tangents and circumcircles, inequalities, harmonic progression, and algebra.

Solve five problems from RMO 2013 and 2014, applying AM-GM inequality, grouping, and case analysis; cover Fermat polynomials, cyclic quadrilaterals, and orthocenter geometry.

Apply modular reasoning and inequalities to solve RMO 2023 problems 111–114, establishing 12 divides abcd, determining triangle perimeters, and using the polynomial trick with x f(x)−abc to find f.

This lecture develops problem solving for math contests by solving four Rmo problems (115–118), proving x rational from x^5 and (20x+19)/x, and examining related inequalities and gp conditions.

Master induction techniques for math contest problems, from base cases and induction steps to strong induction and step-size variations, illustrated with tricky problems like tiling and algebraic cases.

Explore sequences problems using olympiad-style techniques, including induction, telescoping, and inequalities, and apply polynomial and root-of-unity ideas to bound terms and show zero coefficients.