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The measures of the interior angles of a convex hexagon form an increasing arithmetic sequence. How many such sequences are possible if the hexagon is not equiangular and all of the angle degree measures are positive integers less than 150 degrees?
 Sep 5, 2023
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The sum of the interior angles of a hexagon is 180(6−2)=720 degrees. Let the smallest angle in the arithmetic sequence be x degrees. Then the other angles in the sequence are x+d,x+2d,…,x+5d degrees, where d is the common difference. Since the sum of the angles is 720 degrees, we have the equation [x + (x + d) + (x + 2d) + \dots + (x + 5d) = 720.]This is an arithmetic series with first term x and common difference d, so its sum is [\frac{x + (x + 5d)}{2} \cdot 6 = 3x + 15d = 720.]Then x+5d=240. Since x and d are positive integers less than 150, the only possible values of x are 120, 121, 122, …, 148. For each of these values of x, there is exactly one corresponding value of d, namely 240−x. Therefore, there are 29​ possible sequences.

 Sep 5, 2023

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