You can start fire by focusing the sunlight using the magnifying glass. 

I searched the web whether you can do the same using moonlight. And found this and this - the first two in Google search results.

What I found is the thermodynamics argument: you cannot heat anything to a higher temperature using black body radiation than the black body itself, and Moon isn't hot enough.

It may be true, but my gut feelings protest... The larger your aperture is, the more light you collect, also you have better focus because the airy disk is smaller. So if you have a really huge lens with a really short focus (to keep Moon's picture small), or in the extreme case you build a Dyson-sphere around the Moon (letting a small hole to the let the sunlight enter), and focusing all reflected light into a point it should be more than enough to ingnite a piece of paper isn't it? 

I'm confused. So can you start fires using the Moon?

Moonlight has a spectral peak around $650\ \mathrm{nm}$ (the sun peaks at around $550\ \mathrm{nm}$). Ordinary solar cells will work just fine to convert it into electricity. The power of moonlight is about $500\,000$ times less than that of sunlight, which for a solar constant of $1000\ \mathrm{W/m^2}$ leaves us with about $2\ \mathrm{mW/m^2}$. After accounting for optical losses and a typical solar cell efficiency of roughly $20\ \%$, we can probably hope to extract approx. $0.1\ \mathrm{mW}$ with a fairly simple foil mirror of $1\ \mathrm{m^2}$ surface area. Accumulated over the course of a whole night with a full moon, this leaves us with around $6\ \mathrm h\times3600\ \mathrm{s/h}\times0.1\ \mathrm{mW}\approx2\ \mathrm J$ of energy. That's plenty of energy to ignite a fire using the right chemicals and a thin filament as a heater. 