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cut isosceles rigth triangle in 9 triangles (2022 Novosibirsk Oral Geo Oly 8.6)

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September 2, 2023
geometrycombinatorial geometryright triangleisoscelespartitions

Problem Statement

Anton has an isosceles right triangle, which he wants to cut into 99 triangular parts in the way shown in the picture. What is the largest number of the resulting 99 parts that can be equilateral triangles?
A more formal description of partitioning. Let triangle ABCABC be given. We choose two points on its sides so that they go in the order AC1C2BA1A2CB1B2AC_1C_2BA_1A_2CB_1B_2, and no two coincide. In addition, the segments C1A2C_1A_2, A1B2A_1B_2 and B1C2B_1C_2 must intersect at one point. Then the partition is given by segments C1A2C_1A_2, A1B2A_1B_2, B1C2B_1C_2, A1C2A_1C_2, B1A2B_1A_2 and C1B2C_1B_2. https://cdn.artofproblemsolving.com/attachments/0/5/5dd914b987983216342e23460954d46755d351.png
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