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Prove that the greatest solution $n$ of the inequation $c_n<2009$ is a prime

Source: Moldova JTST 2009

May 10, 2023
combinatorics

Problem Statement

Side of an equilatreal triangle has the length n∈N.n\in\mathbb{N}. Each side is divided in n n equal segments by division points. A line parallel with the third side of the triangle is drawn through the division points of every two sides. Let cnc_n be the number of all rhombuses with sidelength 11 inside the initial triangle. Prove that the greatest solution n n of the inequation cn<2009c_n<2009 is a prime number.
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