1997 AIME Problems/Problem 6
Point is in the exterior of the regular -sided polygon , and is an equilateral triangle. What is the largest value of for which , , and are consecutive vertices of a regular polygon?
Let the other regular polygon have sides. Using the interior angle of a regular polygon formula, we have , , and . Since those three angles add up to ,
Clearly is maximized when .
As above, find that using the formula for the interior angle of a polygon.
Solve for to find that . Clearly, for to be positive.
With this restriction of , the larger gets, the smaller the fraction becomes. This can be proven either by calculus, by noting that is a transformed hyperbola, or by dividing out the rational function to get
Either way, minimizng will maximize , and the smallest integer such that is positive is , giving
From the formula for the measure for an individual angle of a regular n-gon, , the measure of . Together with the fact that an equilateral triangle has angles measuring 60 degrees, the measure of (Notice that this value decreases as increases; hence, we are looking for the least possible value of ). For to be vertices of a regular polygon, must be of the form , where is a natural number greater than or equal to 3. It is obvious that . The least angle satisfying this condition is . Equating this with and solving yields
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