Question

圆的一条弦垂直于一条半径，且交点是这条半径的中点。该弦把圆分成两个区域，较大区域面积与较小区域面积的比可表示为 $\frac{a\pi + b\sqrt{c}}{d\pi - e\sqrt{f}}$ ，其中 $a$ ， $b$ ， $c$ ， $d$ ， $e$ ， $f$ 都是正整数， $a$ 与 $e$ 互质，且 $c$ 和 $f$ 都不被任何质数的平方整除。求 $a \cdot b \cdot c \cdot d \cdot e \cdot f$ 除以 $1000$ 的余数。

A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\frac{a\pi + b\sqrt{c}}{d\pi - e\sqrt{f}},$ where $a,$ $b,$ $c,$ $d,$ $e,$ and $f$ are positive integers, $a$ and $e$ are relatively prime, and neither $c$ nor $f$ is divisible by the square of any prime. Find the remainder when the product $a \cdot b \cdot c \cdot d \cdot e \cdot f$ is divided by $1000.$

答案

答案：592

知识点：扇形弦圆面积

难度评级：2050

解答：

按比例缩放，使半径为 $2$ 。弦到圆心的距离为 $1$ ，所以连到弦两端的半径各与被平分的半径成 $60^\circ$ 角，两条端点半径形成的圆心角为 $120^\circ$ 。它们截出的等腰三角形面积为 $\frac{1}{2} \cdot 2 \cdot 2 \sin 120^\circ = \sqrt{3}$ ，整个圆盘面积为 $4\pi$ 。

较小区域是 $120^\circ$ 扇形减去该三角形，面积为 $\frac{4\pi}{3} - \sqrt{3}$ ；较大区域是剩余部分，面积为 $\frac{8\pi}{3} + \sqrt{3}$ 。比值为 $\frac{\frac{8\pi}{3} + \sqrt{3}}{\frac{4\pi}{3} - \sqrt{3}} = \frac{8\pi + 3\sqrt{3}}{4\pi - 3\sqrt{3}},$ 因而 $(a, b, c, d, e, f) = (8, 3, 3, 4, 3, 3)$ 。

乘积为 $8 \cdot 3 \cdot 3 \cdot 4 \cdot 3 \cdot 3 = 2592$ ，除以 $1000$ 的余数为 $592$ 。

Scale so the radius is $2.$ The chord lies at distance $1$ from the center, so each radius to an endpoint of the chord makes a $60^\circ$ angle with the bisected radius, and the two endpoint radii form a central angle of $120^\circ.$ The isosceles triangle they cut off has area $\frac{1}{2} \cdot 2 \cdot 2 \sin 120^\circ = \sqrt{3},$ and the whole disk has area $4\pi.$

The smaller region is the $120^\circ$ sector minus the triangle, $\frac{4\pi}{3} - \sqrt{3},$ and the larger region is the rest, $\frac{8\pi}{3} + \sqrt{3}.$ The ratio is $\frac{\frac{8\pi}{3} + \sqrt{3}}{\frac{4\pi}{3} - \sqrt{3}} = \frac{8\pi + 3\sqrt{3}}{4\pi - 3\sqrt{3}},$ which has the required form with $(a, b, c, d, e, f) = (8, 3, 3, 4, 3, 3).$

The product is $8 \cdot 3 \cdot 3 \cdot 4 \cdot 3 \cdot 3 = 2592,$ whose remainder upon division by $1000$ is $592.$

2004 AIME II 第 1 题

答案：592

难度评级：2050

解答：

其他年份的第 1 题