PUMaC 2020 · 团队赛 · 第 7 题

PUMaC 2020 — Team Round — Problem 7

Let X , Y , and Z be concentric circles with radii 1, 13, and 22, respectively. Draw points A , B , and C on X , Y , and Z , respectively, such that the area of triangle ABC is as large as 2 possible. If the area of the triangle is ∆, find ∆ . 1

解析

Let X , Y , and Z be concentric circles with radii 1, 13, and 22, respectively. Draw points A , B , and C on X , Y , and Z , respectively, such that the area of triangle ABC is as large as 2 possible. If the area of the triangle is ∆, find ∆ . Proposed by: Daniel Carter Answer: 24300 Let the circles be centered at the origin O and without loss of generality A = (1 , 0). Consider fixing A and B and letting C vary. The area of the triangle is maximized when the height from C onto AB is perpendicular to the tangent of Z at C , or in other words when CO is perpendicular to AB . Likewise we have AO is perpendicular to BC , so B and C have the same x -coordinate. Let B = ( x, b ) and C = ( x, c ) with x and b negative and c positive. 2 2 2 2 Then the circle equations give x + b = 169 and x + c = 484, and CO ⊥ AB gives x ( x − 1) + bc = 0. Solve the first two equations for b and c and plug into the third to give √ 2 2 x ( x − 1) + (169 − x )(484 − x ) = 0. Rearranging, squaring, and simplifying gives the cubic 3 2 x − 327 x + 40898 = 0. We know x is negative, so we can look for a root of the form − n 4 2 2 where n is a factor of 40898 = 2 · 11 · 13 . We don’t need to try many to find the solution √ √ √ √ x = − 11. Then b = − 4 3, c = 11 3, and the area of the triangle is 90 3 = 24300.