Combinatorics for Math Contests (AMC, AIME & IMO)

Explore core counting techniques for olympiad problems, including addition and multiplication principles, permutations, combinations, inclusion and exclusion, bijections, and induction.

Explore invariance and monovariants to solve olympiad problems. Learn to use conserved sums, products, parity, and modular invariants with concrete board and token examples.

Explore combinatorial games, apply Zermelo's theorem to two-player, perfect-information play, and learn winning strategies through copycat and reflection techniques in Nim-like and geometric examples.

Explore the pigeonhole principle and apply it to classic contest problems in combinatorics, including friendship counts, parity arguments for lattice points, six-person patterns, sums mod ten, and powers mod 2023.

The lecture introduces key combinatorial identities, including Pascal's triangle relations, hockey stick identities, and Vandermonde's identity, and shows applying them to olympiad problems like IMO 1981 averaging smallest elements.

Learn the principle of inclusion and exclusion to count unions of sets, using add, subtract, and add logic with contest-style problems.

Explore constructive counting, building items step by step while tracking choices at each stage. Apply to palindromes, distinct-digit numbers, racket-signing sequences, and cute six-digit integers, with a preview of bijections.

Learn to count through bijections by establishing a one-to-one and onto mapping between sets, turning difficult counts into easier ones with concrete examples in partitions, divisors, and probability problems.

Explore expected value as a weighted average in Olympiad problems, and apply independence to prove E[X+Y] = E[X] + E[Y].

Explore distributions using stars and bars to count ways to allocate identical items to distinct recipients, with and without restrictions on empty boxes, plus related problem techniques.

Explore the Fibonacci sequence through recursion and recurrence relations, illustrated with counting stairs, no-consecutive-subsets, and no two heads in a row problems, revealing their connection to structured counting.

Derives the general term for Fibonacci numbers using Binet's formula, explores the golden ratio, and introduces Lucas numbers with their own closed form and key identities.

Learn how recurrence relations are formed and solved, using linear recurrences, characteristic equations, and initial values to determine constants, with parking spots and finishing orders examples.

Explore Catalan numbers through non-crossing handshakes, balanced parentheses, and related counting problems, and learn how recursion connects handshake configurations, parentheses, and lattice paths in combinatorics.

Explore Catalan numbers arising from handshake problems, balanced parentheses, and non-crossing lattice paths, and derive their recurrence and closed form via a bijection with paths from (0,0) to (n-1,n+1).

Apply generating functions to convert counting problems into polynomial coefficient extractions, using finite and infinite series, as shown through urns, dice, and money problems, including 4950 ways to distribute chocolates.

Apply generating functions to count partitions of n, derive the product form, and show equivalences such as partitions into even parts and no repeats equaling partitions into only odd parts.

Define graphs, vertices, and edges; distinguish simple and directed graphs, and loops. Analyze degree, multigraphs, complete graphs, and handshake-style proofs about even sums, odd degrees, and shortest paths.

delve into core graph theory concepts, including cycles, paths, simple paths, connected graphs, trees, and Eulerian and Hamiltonian paths and cycles, with problem-based explorations.