# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Just A “Natural” Thing. Topic: Algebra. Level: AIME/Olympiad. April 28th, 2007

Problem: Factor .

Solution: Consider the polynomial . We have

,

so we want to factor the second term when . Call it so that . Consider the relation

.

Since is a root of the LHS, we factor it out of the RHS as well to get

.

Dividing through by and rearranging, we obtain the nice expression

.

Letting , this becomes

which is conveniently enough a difference of two squares. And we’ll leave it as this because the factors are not particularly nice or anything. QED.

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Comment: This “natural” factorization was basically the one crucial step to solving the USAMO #5 this year and most people did not see it, unsurprisingly. Taking the difference was the trickiest/cleverest part, and there were definitely a limited number of approaches to this factorization. Oh well, at least it seems sort of cool after you know about it.

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Practice Problem: (2007 USAMO – #5) Prove that for every nonnegative integer n, the number is the product of at least (not necessarily distinct) primes.

### One Response to “Just A “Natural” Thing. Topic: Algebra. Level: AIME/Olympiad.”

1. t0rajir0u Says:

Man, I wish I could come up with problems this contrived.