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Washington and Lee University

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April Problems

Problem 1 (not too hard):  Define  f1 = (1 - x)/(x + 1), and define fn(x) = fn-1(f1(x)). In other words, f2(x) = f1(f1(x)), f3 = f1(f1(f1(x))), and so on. Find f2010(2010).

Problem 2 (a bit tricky): Find all triples (x,y,z) such that when any one of these three numbers is added to the product of the other two, then the result is 2.  Be sure to prove Be sure to prove (or write a convincing
argument) that you've found all of them!

Problem 3 (somewhat hard):  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. Show that it is possible to make the calculator display 2 by pressing a finite sequence of the above six buttons. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians. (Hint: you might want to determine various formulas such as  sin(arccos(x)) or  tan(arcsin(cos(arctan(x)))). A calculus book would be helpful here.)