PUMaC 2023 · 团队赛 · 第 8 题

PUMaC 2023 — Team Round — Problem 8

Let △ ABC be a triangle with sidelengths AB = 5, BC = 7, and CA = 6. Let D, E, F be the feet of the altitudes from A, B, C , respectively. Let L, M, N be the midpoints of sides BC, CA, AB , respectively. If the area of the convex hexagon with vertices at D, E, F, L, M, N √ x y can be written as for positive integers x, y, z with gcd( x, z ) = 1 and y square-free, find z x + y + z . 4 3 2

解析

Let △ ABC be a triangle with sidelengths AB = 5, BC = 7, and CA = 6. Let D, E, F be the feet of the altitudes from A, B, C , respectively. Let L, M, N be the midpoints of sides BC, CA, AB , respectively. If the area of the convex hexagon with vertices at D, E, F, L, M, N √ x y can be written as for positive integers x, y, z with gcd( x, z ) = 1 and y square-free, find z x + y + z . Proposed by Sunay Joshi Answer: 10043 Let [ P ] denote the area of polygon P . Let a, b, c denote BC, CA, AB , respectively. First note that the correct ordering of the vertices of the convex hexagon in counterclockwise order is DN F EM L . Therefore we have the identity [ DN F EM L ] = [ ABC ] − [ BN D ] − [ CLM ] − [ AEF ] 1 Since CLM is similar to ABC with CL/CB = 1 / 2, we have [ CLM ] = [ ABC ]. Also, since 4 2 AEF is similar to ABC with AE/AB = cos A , we have [ AEF ] = cos A · [ ABC ]. Finally, 1 1 BD c cos B [ BN D ] = [ ABD ] = [ ABC ] = · [ ABC ] 2 2 BC 2 a √ √ 2 2 2 b + c − a By Heron’s Formula, [ ABC ] = 9 · 4 · 3 · 2 = 6 6. By the Law of Cosines, cos A = = 2 bc 2 2 2 1 a + c − b 19 and cos B = = . Plugging these values in, we find 5 2 ac 35 √ √ 19 1 1 7587 6 [ DN F EM L ] = 6 6 · 1 − − − = 98 4 25 2450 Our answer is therefore 7587 + 6 + 2450 = 10043. 4 3 2