1985 AHSME Problems/Problem 14
Problem
Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon?
Solution
Suppose that such a polygon has sides. Let the three obtuse angle measures, in degrees, be , , and and the acute angle measures, again in degrees, be .
Since for each , we have and similarly, since for each , It follows that and recalling that the sum of the interior angle measures of an -gon is , this reduces to . Hence so an upper bound is , and it is easy to check that this bound can be attained by e.g. a convex hexagon with a right angle, acute angles, and obtuse angles, as shown below:
Accordingly, the maximum possible number of sides of such a polygon is .
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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