Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.5 AIME, harder Prove it JBMO

Problem:
Can we divide an equilateral triangle $\triangle ABC$ into 2011 small triangles using 122 straight lines? (There should be 2011 triangles that are not themselves divided into smaller parts and there should be no polygons which are not triangles.)

Solution

Solution:
Firstly, for each side of the triangle, we draw 37 equidistant, parallel lines to it. In this way we get $38^{2} = 1444$ triangles.

Then we erase 11 lines which are closest to the vertex $A$ and parallel to the side $BC$ and we draw 21 lines perpendicular to $BC$ , the first starting from the vertex $A$ and 10 on each of the two sides, the lines which are closest to the vertex $A$ , distributed symmetrically. In this way we get $26 \cdot 21 + 10 = 556$ new triangles.

Therefore we obtain a total of 2000 triangles and we have used $37 \cdot 3 - 11 + 21 = 121$ lines.

Let $D$ be the $12^{\text{th}}$ point on side $AB$ , starting from $B$ (including it). The perpendicular to $BC$ passing through $D$ will be the last line we draw. In this way we obtain the required configuration.

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