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Geometry Difficulty 4.7 AIME Prove it United States

Problem:

Consider an equilateral triangle $T$ of side length $12$ . Matthew cuts $T$ into $N$ smaller equilateral triangles, each of which has side length $1$ , $3$ , or $8$ . Compute the minimum possible value of $N$ .

Solution

Solution:

Matthew can cut $T$ into $16$ equilateral triangles with side length $3$ . If he instead included a triangle of side $8$ , then let him include $a$ triangles of side length $3$ . He must include $12^{2} - 8^{2} - 3^{2} a = 80 - 9a$ triangles of side length $1$ . Thus $a \leq 8$ , giving that he includes at least
$(80 - 9a) + (a) + 1 = 81 - 8a \geq 17$
total triangles, so $16$ is minimal.

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