# Mathematical Food for Thought

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

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Function + Equation = ?? Topic: Algebra. Level: AIME/Olympiad. August 22nd, 2006

Problem: Find all functions that satisfy for all reals .

Solution: Let’s start by plugging in something easy, like . We get

so . Hmm, make it a little more complicated then and try just . Then

.

But wait, we already know so we can just plug this in. Thus we have

or .

We check by plugging each in:

(this works)

(this doesn’t).

That means only works. QED.

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Comment: Functional equations are pretty strange as far as problems go. Basically you go around plugging random things in until you find something useful. Then you work it all out and you usually get an interesting result.

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Practice Problem: (360 Mathematical Contests – 1.1.49) Find all polynomials with integral coefficients such that

for all real numbers .

### 5 Responses to “Function + Equation = ?? Topic: Algebra. Level: AIME/Olympiad.”

Just a note to commentors: Don’t put links in your comments. Spammers have been posting on my blog lately, so if you put a link it most likely will get spam-blocked.

2. sup Says:

P(x) = x

analyze degree and leading coefficient to see deg P = 1, then its EZ

OK, I think this is the best way to do this; if you want to post a comment, just register quickly and it should work fine.

4. idiot Says:

I got an account!!!! great.. now i’m referred to as the idiot.
do you happen to be able to know all the passwords?? just curious