HMMT 二月 2025 · 冲刺赛 · 第 12 题

HMMT February 2025 — Guts Round — Problem 12

[7] Holden has a collection of polygons. He writes down a list containing the measure of each interior ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ angle of each of his polygons. He writes down the list 30 , 50 , 60 , 70 , 90 , 100 , 120 , 160 , and x , in some order. Compute x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2025, February 15, 2025 — GUTS ROUND Organization Team Team ID#

解析

[7] Holden has a collection of polygons. He writes down a list containing the measure of each interior ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ angle of each of his polygons. He writes down the list 30 , 50 , 60 , 70 , 90 , 100 , 120 , 160 , and ◦ x , in some order. Compute x . Proposed by: Rishabh Das Answer: 220 Solution: We work in degrees. The sum of all 9 angles is 680 + x . The sum of the angles in a polygon with n sides is 180( n − 2) ≡ 180 n mod 360. Since there are 9 angles, the polygons have a total of 9 sides, so the sum of the 9 angles must be 9 · 180 ≡ 180 mod 360. Thus 680 + x ≡ 180 mod 360, so x ≡ 220 mod 360. Since 0 < x < 360, we know x = 220 .