### An exercise in transfinite magic

In my hometown there is a university called SUSU. And in this university regular math contests (link in Russian) are held, organized by A. Yu. Evnin. The following problem appeared in one of these contests:

**Problem 1.** A function has the following property: every straight line on the plane intersects the graph of and the parabola at the same number of points. Prove that for every .

The solution is quite straightforward, I am not going to write it down. It is probably enough to say that it heavily exploits the convexity of .

This post is not about problem 1. It is about the next thing that comes to mind:

**Problem 2.** Two functions have graphs and such that for every straight line : . Does it follow that and are the same function?

If you think about this one for a minute, I’m sure you’ll find it much more challenging than problem 1. This whole post is about solving problem 2. If you want to give it a try yourself, this is the last point in the text where you can still do this without knowing the answer.