5 link to gauss-lucas theorem didn't work for me so i fixed it

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

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

Thank you.

Post Closed as "off topic" by Pete L. Clark, Qiaochu Yuan, Mariano Suárez-Alvarez, Andrew Stacey, Andy Putman
4 edited body

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

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

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

Thank you.

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