Unpopular "elementary" theorems/identities to impress an audience of mathematicians.

2011-01-03T13:31:00Z

This question grew out of my recent job interview. Since the interviewers were math professors, I had a hard time searching for interesting elementary theorems in case I got asked for one. I thought topics such as the Banach-Tarski paradox, Godel's theorems, the Mandelbrot set, the Brouwer Fixed Point Theorem, etc were well-known and wouldn't do the job. However, after a cursory search, I found some to my taste:

1.~~Marden's theorem (or here)(It is not Marsden.)~~ Gauss–Lucas theorem

2.The identity $\int_{0}^1 \frac{x^4(1-x)^4}{1+ x^4} dx = \frac{22}{7}- \pi$

So, my question here is an invitation to expand the list (of theorems that would get an interviewee accepted).

To recap, my criteria for selection are

Not widely known,
Elementary- understandable to a first year grad student, and
Interesting-i.e. MOtizens, assuming they are the audience, will be delighted to have come across it.

Thank you.

Answer by Andrey Rekalo for Unpopular "elementary" theorems/identities to impress an audience of mathematicians.

2011-01-03T13:43:32Z

The AM-GM inequality is implied by an identity.

For a function $f=f(x_1,x_2,\dots,x_n):\mathbb R^n\to\mathbb R$ let $Pf(x_1,x_2,\dots,x_n)$ denote the sum of $f$ over the $n!$ quantities that result from all possible $n!$ permutations of the $x_i$. Then $$\frac{x_1^n+x_2^n+\dots+x_n^n}{n}-x_1x_2\dots x_n=\frac{1}{2\ n!}(\phi_1+\phi_2+ \dots \phi_n),$$ where $$\phi_k=P[(x_1^{n-k}-x_2^{n-k})(x_1-x_2)x_3x_4\dots x_{k+1}]$$ $$=P[(x_1-x_2)^2(x_1^{n-k-1}+\dots x_2^{n-k-1})x_3x_4\dots x_{k+1}]\geq0.$$ The proof dates back to Hurwitz.

Unpopular "elementary" theorems/identities to impress an audience of mathematicians. - MathOverflow [closed]

Unpopular "elementary" theorems/identities to impress an audience of mathematicians.

Answer by Andrey Rekalo for Unpopular "elementary" theorems/identities to impress an audience of mathematicians.