I'll describe a problem from William Lowell Putnam Mathematical Competition (54th, problem A1):(adsbygoogle = window.adsbygoogle || []).push({});

There is an image of the graph y = sin x where x ranges from 0 to pi. The graph of y = k where 0 < k < 1 is also drawn there.

Clearly y = k intersects y = sin x at two places, call them (x1, k) and (x2, k).

Let us define

A1 = area bounded above by y = k, below by y = sin x, left by x = 0, and right by x = x1

A2 = area bounded above by y = sin x, below by y = k (the left endpoint is at x = x1 and the right endpoint is at x = x2)

Now we must find k (NOTE by me: the problem asks us to find, not approximate) such that that A1 = A2.

I reduced this problem to the equation:

[tex]

\sin{x_1} = \frac{2}{\pi - 2 + 2x_1}

[/tex]

Where k = sin x1. (try to derive this equation :) )

I can approximate x1, and thus k, using a graphing calculator (and I think it's possible to approximate it using newton's method) but I can't see how I can find the exact value of k. Is it even possible? If so, how?

Thank you.

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# Solving a Putnam Equation

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