HMMT 十一月 2009 · 冲刺赛 · 第 8 题

HMMT November 2009 — Guts Round — Problem 8

[ 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

[ 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 exterior angle 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, 2 so n ≤ 27.