Competition problem — Geometry — Maths Olympiad Prep

Library / /2 of 16

Geometry Difficulty 4.6 AIME Prove it JBMO

Problem:
A triangle with area $2003$ is divided into non-overlapping small triangles. The number of all the vertices of all those triangles is $2005$ . Show that at most one of the smaller triangles has area less or equal to $1$ .

Solution

Solution:
Since all the vertices are $2005$ , and the vertices of the big triangle are among them, it follows that the number of the small triangles is at least $2003$ . So, it follows that at least one of the small triangles has area at most $1$ .

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

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