PUMaC 2022 · 几何（B 组） · 第 4 题

PUMaC 2022 — Geometry (Division B) — Problem 4

An ellipse has foci A and B and has the property that there is some point C on the ellipse such that the area of the circle passing through A , B , and, C is equal to the area of the ellipse. √ a + b 2 Let e be the largest possible eccentricity of the ellipse. One may write e as , where a, b, c and c are integers such that a and c are relatively prime, and b is not divisible by the square 2 2 2 of any prime. Find a + b + c .

解析

An ellipse has foci A and B and has the property that there is some point C on the ellipse such that the area of the circle passing through A , B , and, C is equal to the area of the ellipse. √ a + b 2 Let e be the largest possible eccentricity of the ellipse. One may write e as , where a, b, c and c are integers such that a and c are relatively prime, and b is not divisible by the square 2 2 2 of any prime. Find a + b + c . Proposed by Daniel Carter Answer: 30 Consider the ellipse with largest possible eccentricity that has this property. The smallest possible area of the circle is when the center of the circle is the center of the ellipse. Let O be 2 the center of the ellipse. Then π ( OA ) = πRr , where R, r are the semi-major and semi-minor 2 2 2 axes. We have OA/R = e , so then ( OA ) e = r . Noting that r = R − ( OA ) , this means √ − 1+ 5 2 2 2 2 2 2 e = (1 /e − 1), or e = . So the answer is ( − 1) + 5 + 2 = 30. 2