A calculator is broken so that the only keys that still work are the sin, cos, tan, arcsin, arccos, and arctan buttons. The display initially shows 0. Given any positive rational number q, show that pressing some finite sequence of buttons will yield q. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.
I’ve started reading Zeitz’s The Art and Craft of Problem Solving. This one took me about 90 minutes, though as usual, once I had a solution it seemed obvious. Originally from USAMO 1995. Who comes up with these problems? How? You sit down and say, “Okay, it’s time to invent a problem that can be solved with elementary math, but only if you see some diabolical trick”, and then you do what?