I found the following problem in a captcha: What does it mean, and what would the solution be ?

First, we have $$ \begin{align} \frac1{z^4+z^2+1} &=\frac1{12}\left( \frac{-3-i\sqrt3}{z-e^{\pi i/3}} +\frac{3+i\sqrt3}{z-e^{4\pi i/3}} +\frac{3-i\sqrt3}{z-e^{2\pi i/3}} +\frac{-3+i\sqrt3}{z-e^{5\pi i/3}} \right)\tag{1} \end{align} $$ Let $\gamma$ be the rectangle $$ [-1-i,1-i]\cup[1-i,1+i]\cup[1...

When I was a child I was given this problem to send a wire from electricity, water, and internet to each of the houses, all three houses must have all three wires connected without being crossed over each other (wires can't meet and there is no such thing as a wire going on top of another wire)...

When I showed to my brother how I proved \begin{equation} \int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2 \end{equation} using the following theorem by Mr. Olivier Oloa \begin{equation}{\large\int_{0}^{\!\Large \frac{\pi}{2}}} \frac{\cos \left(\! s...