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, 2019

Geometry Difficulty 5.1 AIME, harder Prove it United States

Problem:

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 $d$ ?

Solution

Solution:

The sum of the internal angles of a quadrilateral is $360^{\circ}$ . 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, $d \geq 360^{\circ} / 3 = 120$ . It is not difficult to see that for any $0 < \alpha < 120$ , a quadrilateral of which three angles have measure $\alpha$ degrees and the fourth angle has measure $(360 - 3\alpha)$ degrees can be constructed.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.