Season 3 Problem Posts
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Problem Post 3-1: How many Shapes, Part 4: Triangles - We conclude our tour of shape counting with something unexpectedly nontrivial: counting triangles!
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Season 2 Problem Posts
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Factor Sums and the Distributive Law - A little insight into the distributive property can go a long way into counting and summing divisors.
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A Sampler of `Olympiad' Geometry Concepts - This post is meant to be a `sampler' of sorts, to show the most common tag words one will see in olympiad geometry problems. We solve two problems in such a way as to form a whirlwind tour of the subject.
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On the Escalation of Packing Problems - How many rectangular boxes fit into a fixed rectangular pallet? This is supposed to be an easy question, or is it?
As it turns out, change the numbers ever so slightly, and we could effectively turn a fifteen-second exercise to a research-level Godzilla.
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EROS should be Taught in High School - Some very basic linear algebra can be employed fruitfully at the high school level.
Never get muddled with systems of linear equations again!
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How many Shapes, Part 2: Rectangles in Rectangles - A simple Internet puzzle escalates into a frenzy of rectangle counting in different scenarios.
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How many Shapes, Part 3: Rectangles in Not-Rectangles - How about overlapping rectangles and staircase figures? It's a simple idea, but the execution startles.
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Season 1 Problem Posts
- Kookie's Cookies [PDF] - We begin Project Phi (then known as Solvespace) with a variant of the famous Josephus problem.
- Ellipse Tracing [PDF] - Far as I know, contests don't usually serve problems that treat ellipses as the geometric figures that they are (often, they are viewed algebraically). What happens when they do?
- Coin Rotation [PDF] - A favorite among grade school contests, this problem banks on the multiple rotational components arising when round objects are rolled about other round objects.
- Mass Points - "Give me a place to stand, and I will move the Earth." The unorthodox method of mass points can halve solving time for your usual area chasing problem.
- Trivial Inequality [PDF] - Sometimes the most menacing algebraic expression is simply a big wimpy square. The rest is trivial.
- Romance of a Right Triangle [PDF] - Our first Level 4 problem from the CMO, where we take a wild romp chasing angles and similarities around a right triangle.
- Back-Engineering [PDF] - We're often made to find, say, the number of diagonals of an $n$-gon. What happens if we're asked to do the reverse?
- Meticulous Manipulations [PDF] - Almost everybody hates having to deal with cross terms like $xy$.
- Triangle Side Expression [PDF] - Trying to prove symmetric inequalities about triangle sides can get nasty, unless you apply some transformations first.
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