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Difficulty: Very hardPolygon Angles and Properties

An irregular convex polygon has nn sides. The measures of its interior angles, in degrees, are all distinct integers. If all of the interior angles are obtuse, what is the maximum possible value of nn?

Answer: 26

Answer

The maximum possible value of nn is 26.
For a convex 2626-gon, the sum of the interior angles is 24×180∘=4320∘24 \times 180^\circ = 4320^\circ. We can choose 2626 distinct integer angles from the range [91∘,179∘][91^\circ, 179^\circ] that sum to exactly 4320∘4320^\circ because the maximum possible sum of 2626 distinct integers in this range is 4329∘4329^\circ, which is greater than 4320∘4320^\circ. For n=27n = 27, the sum of the interior angles must be 25×180∘=4500∘25 \times 180^\circ = 4500^\circ, but the maximum possible sum of 2727 distinct integers in the range is only 4482∘4482^\circ, which is less than 4500∘4500^\circ. Therefore, 2626 is the maximum value of nn.

Step-by-Step Solution

1
Determine the set of possible angle measures.
The angles must be integers in the range [91∘,179∘][91^\circ, 179^\circ].
Interior angles of a convex polygon must be less than 180∘180^\circ. Since they are obtuse and distinct integers, they must be strictly greater than 90∘90^\circ, giving the range [91,179][91, 179].
2
Write the sum of the interior angles of a convex nn-gon.
Sum =(n−2)×180∘= (n - 2) \times 180^\circ.
This is the standard formula for the sum of the interior angles of any convex nn-gon.
3
Find the maximum possible sum of nn distinct angles in the range [91,179][91, 179].
Maximum Sum =179n−n(n−1)2= 179n - \frac{n(n - 1)}{2}.
The maximum sum is achieved by selecting the largest nn integers from the set: 179,178,…,179−(n−1)179, 178, \dots, 179 - (n - 1).
4
Set up the inequality and simplify.
n2+n−720≤0n^2 + n - 720 \le 0.
Since the sum of the angles must be less than or equal to the maximum possible sum, we have (n−2)×180≤179n−n(n−1)2(n-2) \times 180 \le 179n - \frac{n(n-1)}{2}. Multiplying by 2 and simplifying yields the quadratic inequality.
5
Solve the quadratic inequality for the largest integer nn.
n=26n = 26.
Evaluating the quadratic expression for consecutive integers: for n=26n = 26, 262+26−720=−18≤026^2 + 26 - 720 = -18 \le 0; for n=27n = 27, 272+27−720=36>027^2 + 27 - 720 = 36 > 0. Thus, 26 is the maximum possible integer value.

Key Concept

Sum of interior angles of a convex polygon combined with algebraic optimization.
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Topics

Polygon Angles and Properties

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