Sample · 35-Olympiad Strategy and Advanced Problem Solving
Chapter 35 Overview
Book 7 · Advanced Olympiad Problem Solving · CHAPTER 35: Olympiad Strategy and Advanced Problem Solving
The final chapter is not about a new theorem — it is about how the strongest solvers actually think. Two contestants can know the same facts, yet one cracks a problem in twenty minutes and the other stares for two hours. The difference is strategy: knowing which move to try, how to structure a proof, and how to spend a limited clock. This chapter distills the master skills: introducing an auxiliary construction to unlock a problem, proving upper and lower bounds to pin down extremes, exploiting symmetry and WLOG to shrink the work, reverse-engineering from equality cases, discovering the right lemmas, weighing multiple methods, managing contest time, and writing a rigorous final solution. We keep it lively with comics, map every method with a flowchart, draw plenty of figures, and ground each idea in real contest practice. Ten Topics — 35.1 Auxiliary Construction, 35.2 Upper Bounds, 35.3 Lower Bounds, 35.4 Symmetry, 35.5 WLOG, 35.6 Equality Cases, 35.7 Lemma Discovery, 35.8 Multiple Methods, 35.9 Time Management, 35.10 Solution Writing — run at HMMT-to-IMO level, each with worked reasoning, contest-craft tips, figures, and eight fully solved questions. Solve like a champion!
35.1 Auxiliary Construction
35.1 Auxiliary Construction
An auxiliary construction is a new object — a point, a line, a circle, a variable, or a cleverly chosen expression — that you add to a problem to reveal hidden structure. The given configuration may hide a right angle, a similarity, or a telescoping sum until the extra object makes it visible. In geometry this is the famous “draw the auxiliary line”: connect two points, drop a perpendicular, extend a segment, or reflect a vertex, and suddenly congruent triangles or a cyclic quadrilateral appear.
The same move works far beyond geometry. In algebra you introduce a variable for an unknown quantity or a symmetric substitution; in combinatorics you build an auxiliary graph or a bijection; in inequalities you add and subtract a term to complete a square. The art is choosing a construction that connects the given data to the goal — a good auxiliary object turns an opaque problem into a routine one. When stuck, ask: what single new object would let a known theorem apply?
📋 Auxiliary constructions
| Setting | Typical construction |
|---|---|
| geometry | auxiliary line, perpendicular, reflection, circle |
| algebra | introduce a variable or substitution |
| inequalities | add and subtract a term; complete a square |
| combinatorics | auxiliary graph or bijection |
🌍 Real-World Application — adding a helper variable in modeling
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