Cutting polygons in half

This Riddler puzzle is about cutting polygons in half. Here is the problem:

The archvillain Laser Larry threatens to imminently zap Riddler Headquarters (which, seen from above, is shaped like a regular pentagon with no courtyard or other funny business). He plans to do it with a high-powered, vertical planar ray that will slice the building exactly in half by area, as seen from above. The building is quickly evacuated, but not before in-house mathematicians move the most sensitive riddling equipment out of the places in the building that have an extra high risk of getting zapped.

Where are those places, and how much riskier are they than the safest spots? (It’s fine to describe those places qualitatively.)

Extra credit: Get quantitative! Seen from above, how many high-risk points are there? If there are infinitely many, what is their total area?

Here is my solution:
[Show Solution]

And here is a bonus interactive graphic showing the solution

A clever integral

I was recently reminded of this problem from one of my favorite books: Problem-Solving Through Problems. The problem originally appeared in the 1980 Putnam Competition.

Evaluate the following definite integral.

\[
\int_0^{\pi/2} \frac{\mathrm{d}x}{1 + (\tan x)^{\sqrt{2}}}
\]

The solution:
[Show Solution]