# Problem A. 380. (October 2005)

**A. 380.** The convex *n*-sided polygon *K* lies in the interior of a unit square. Show that it is possible to select three vertices of the polygon that form a triangle of smaller area than units.

(5 pont)

**Deadline expired on November 15, 2005.**

**Solution. **We prove that there exist two adjacent sides of *K* which form a sufficiently small triangle.

*K* is convex and is a subset of the unit square. Therefore its perimeter *p* is at most 4.

Denote the length of the *i*th side of *K* by *d*_{i} and let the angle between the *i*th and (*i*+1)th sides be -_{i}. Since

there exists an index *i* for which .

Consider the triangle determnined by the *i*th and (*i*+1)th sides; its area is

### Statistics:

15 students sent a solution. 5 points: Dücső Márton, Erdélyi Márton, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Nagy 224 Csaba, Paulin Roland, Tomon István. 4 points: Bogár 560 Péter. 1 point: 3 students. 0 point: 1 student.

Problems in Mathematics of KöMaL, October 2005