HMMT 二月 2019 · 几何 · 第 1 题

HMMT February 2019 — Geometry — Problem 1

Let d be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than d degrees. What is the minimum possible value for d ?

解析

Let d be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than d degrees. What is the minimum possible value for d ? Proposed by: James Lin Answer: 120 ◦ The sum of the internal angles of a quadrilateral triangle is 360 . To find the minimum d , we note the limiting case where three of the angles have measure d and the remaining angle has measure ◦ approaching zero. Hence, d ≥ 360 / 3 = 120. It is not difficult to see that for any 0 < α < 120, a quadrilateral of which three angles have measure α degrees and fourth angle has measure (360 − 3 α ) degrees can be constructed.