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Optimization: Squares, Constraints, and Symmetry

Fall 2026 · 9 problems

Completing squares, dot products and symmetry find exact maxima and minima without multivariable calculus. Five warm-ups lead to three Putnam problems, ending with the largest circle that fits inside a four-dimensional cube (Putnam 2008 B3).

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What is in it

Problems only — the club works these out together at meetings. Topic and attribution are listed as a starting point.

1. A Quadratic with a Floor Completing SquaresEasy
2. A Plane and a Sphere Dot ProductsEasy
3. The Range of a Moving Coordinate TrigonometryEasy
4. A Quadratic on a Circle Quadratic FormsEasy–Medium
5. A Circle with Many Coordinates Vectors / Parametric GeometryMedium
6. A Cube in a Cone Geometry / Putnam First PositionPutnam 1998 A1
7. Packing Two Squares Optimization / Putnam First PositionPutnam 1996 A1
8. Three Coordinates, One Product Optimization / Competition Style
9. A Circle in a Four-Dimensional Cube Geometric Optimization / Putnam B3Putnam 2008 B3