1995 USAMO Problems/Problem 2

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Problem

A calculator is broken so that the only keys that still work are the $\, \sin, \; \cos,$ $\tan, \; \sin^{-1}, \; \cos^{-1}, \,$ and $\, \tan^{-1} \,$ 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.

Solution

We will prove the following, stronger statement : If $m$ and $n$ are relatively prime nonnegative integers such that $n>0$ , then the some finite sequence of buttons will yield $\sqrt{m/n}$ .

To prove this statement, we induct strongly on $m+n$ . For our base case, $m+n=1$ , we have $n=1$ and $m=0$ , and $\sqrt{m/n} = 0$ , which is initially shown on the screen. For the inductive step, we consider separately the cases $m=0$ , $0<m\le n$ , and $n<m$ .