HMMT 十一月 2009 · 冲刺赛 · 第 8 题
HMMT November 2009 — Guts Round — Problem 8
题目详情
英文原题
- [ 7 ] The angles of a convex n -sided polygon form an arithmetic progression whose common difference
(in degrees) is a non-zero integer. Find the largest possible value of n for which this is possible. (A
°
polygon is convex if its interior angles are all less than 180 .)
解析
英文解析
- [ 7 ] The angles of a convex n -sided polygon form an arithmetic progression whose common difference
(in degrees) is a non-zero integer. Find the largest possible value of n for which this is possible. (A
°
polygon is convex if its interior angles are all less than 180 .)
°
Answer: 27 The exterior angles form an arithmetic sequence too (since they are each 180 minus
°
the corresponding interior angle). The sum of this sequence must be 360 . Let the smallest exteriorangle be x and the common difference be d . The sum of the exterior angles is then x + ( x + a ) + ( x +
n ( n − 1)
2 a ) + . . . + ( x + ( n − 1) a ) = · a + nx . Setting this to 360, and using nx > 0, we get n ( n − 1) < 720,
so n ≤ 27.2