"Here's a cool problem": a collection of short questions with clever solutions
Solution 1:
Evaluate:
$$\int_0^1\!\!\int_0^1\! \dfrac{1}{1-xy} \, \mathrm{d}x\mathrm{d}y$$
Solution 2:
Consider a 8x8 chessboard and 2x1 dominos. Clearly one can easily cover the whole chessboard with 32 dominos.
Now omit the squares on the bottom-left and top-right. There are 62 squares left on the chessboard. Can you cover them using 31 dominos?