Research task (STEM):
write a series of blog posts on important combinatorics topics in math olympiads. Include intuitive and detailed explanations, as well as example problems from top olympiads such as USAMO, USA TSTST, CMO, IMO, IMO shortlist, etc. Provide correct solutions for those problems and the insight or takeaways from those solutions.

Evaluation rubric:
criterion: The series covers at least three distinct topics from the following list: Enumerative combinatorics (e.g. generating functions, inclusion-exclusion, combination, permutations), Graph Theory (e.g. Ramsey theory, planar graphs, graph coloring, domino tiling), or Probabilistic Methods (e.g. random graphs, sum-free sets, local lemma, deterministic theorems).
weight: 4
axis: Explicit Criteria

criterion: Each post includes an intuitive and detailed explanation of its core combinatorics topic (e.g. providing baseline definitions, using analogies, leveraging storytelling, visual explanations).
weight: 4
axis: Implicit Criteria

criterion: Formal mathematical definitions and theorems are presented accurately for each topic (e.g. structured presentation, proper notation, formatting). 
weight: 5
axis: Explicit Criteria

criterion: All example problems are clearly labeled with their originating competition and year (e.g. USAMO 2021 P3, IMO 2021 P2, 2015 CMO P4, 2012 USAMO P3).
weight: 3
axis: References & Citation Quality

criterion: Correct and complete solutions with detailed thought process and reasoning behind each step are provided for every example problem.
weight: 5
axis: Explicit Criteria

criterion: For at least one solved problem, the insights section describes how the core technique is adapted to solve a problem from a different mathematical domain (e.g. applying a graph theory concept to a number theory problem, using polynomial roots and properties of algebraic equations on geometric problems, using properties of prime numbers to define relationships between vertices, using concepts from real or complex analysis on distribution of prime numbers) and explicitly outlines the necessary modifications.
weight: 1
axis: Implicit Criteria

criterion: LaTeX formatting is used for all mathematical expressions and equations within the text.
weight: 1
axis: Communication Quality

criterion: Each blog post is written in blog style; it starts with an introduction capturing reader's attention and describes key concepts that are essential for solving example problems.
weight: 4
axis: Instruction Following

criterion: Each blog post articulates at least two distinct problem-solving heuristics (e.g. consider extreme cases and look for variants, draw a diagram and visualize, work backward from the goal, solve by contradiction) and for each, it not only explains its application in the given solution but also describes the potential pitfalls or common misapplications of that heuristic.
weight: 3
axis: Synthesis of Information

criterion: Each blog post contains at least two example problems, and all examples are selected from top math olympiads (e.g. USAMO, USA TSTST, CMO, IMO, IMO Shortlist).
weight: 5
axis: Explicit Criteria

criterion: The response includes visual aids (e.g. diagram, table, graph, chart) if the topic naturally lends itself to it (e.g. graph theory, decision trees, catalan numbers, venn diagrams).
weight: 1
axis: Implicit Criteria

criterion: Each blog post explicitly states the assumed prerequisite mathematical knowledge (e.g. modular arithmetic for complex Pigeonhole Principle problems, counting principles and factorials for permutation problems).
weight: 2
axis: Implicit Criteria

criterion: Alternative solution approaches or viewpoints are discussed for at least one example problem (e.g. solving a combinatorial problem with 1. inclusion-exclusion principle and 2. complementary counting principle).
weight: 3
axis: Implicit Criteria

criterion: Underlying mathematical principles that are beyond the core topic are explicitly identified if relevant (e.g. discussing how set theory can be used for inclusion-exclusion principle, stating use of recursion for catalan number problems).
weight: 1
axis: Implicit Criteria

criterion: The series of posts demonstrates a clear pedagogical progression, where concepts from earlier posts are explicitly referenced and built upon in later posts to solve more complex, multi-part problems (e.g. using an inclusion-exclusion argument on a structure defined by a graph, applying linearity of expectation to find the expected size of a sum-free set, combining local lemma with a random graph, employing a geometric interpretation of algebraic roots).
weight: 2
axis: Synthesis of Information

criterion: The response justifies why the chosen topics are foundational for olympiad-level problem solving (e.g. appears frequently, conceptually simple yet requires exceptional creativity).
weight: 3
axis: Implicit Criteria

criterion: Each post discusses key variations of principles used within the topic (e.g. Pigeonhole general form, strong form, probabilistic inclusion-exclusion, incusion-exclusion on multisets).
weight: 3
axis: Synthesis of Information

criterion: Each blog post contains at least two insights or takeaways from example problems and solutions (e.g. type of problems that appear, key techniques used, strategies often used to tackle problems, ways to think about alternative solutions).
weight: 4
axis: Synthesis of Information

criterion: The response contains insights and takeaways for all examples, not per each topic or each example.
weight: -2
axis: Instruction Following

criterion: The response contains a topic that is not covered by the math olympiad competitions (e.g. advanced calculus, topology, abstract algebra, category theory).
weight: -4
axis: Instruction Following