Maths Olympiad Prep

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Geometry Difficulty 5.2 AIME, harder Find the answer

Can we divide an equilateral triangle ABC\triangle A B C 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)

A number or a short expression. Spacing and $ signs are ignored.

Solution

Firstly, for each side of the triangle, we draw 37 equidistant, parallel lines to it. In this way we get 382=144438^{2}=1444 triangles. Then we erase 11 lines which are closest to the vertex AA and parallel to the side BCB C and we draw 21 lines perpendicular to BCB C, the first starting from the vertex AA and 10 on each of the two sides, the lines which are closest to the vertex AA, distributed symmetrically. In this way we get 2621+10=26 \cdot 21+10= 556 new triangles. Therefore we obtain a total of 2000 triangles and we have used 37311+21=12137 \cdot 3-11+21=121 lines. Let DD be the 12th 12^{\text {th }} point on side ABA B, starting from BB (including it). The perpendicular to BCB C passing through DD will be the last line we draw. In this way we obtain the required configuration.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.