Question

四个全等的半圆内接于边长为 $1$ 的正方形中，使得它们的直径在正方形的边上，每条直径的一个端点位于正方形的一个顶点，并且相邻半圆互相相切。一个以正方形中心为圆心的小圆与四个半圆都相切，如下图所示。

小圆的直径可以写成 $(\sqrt{a} + b)(\sqrt{c} + d)$ ，其中 $a, b, c$ 和 $d$ 是整数。求 $a + b + c + d$ 。

Four congruent semicircles are inscribed in a square of side length $1$ so that their diameters are on the sides of the square, one endpoint of each diameter is at a vertex of the square, and adjacent semicircles are tangent to each other. A small circle centered at the center of the square is tangent to each of the four semicircles, as shown below.

The diameter of the small circle can be written as $(\sqrt{a} + b)(\sqrt{c} + d),$ where $a, b, c,$ and $d$ are integers. What is $a + b + c + d?$

$3$

$5$

$8$

$9$

$11$

答案：A

知识点：相切圆坐标几何根式

难度评级：1930

解答：

设每个半圆半径为 $\rho$ ，圆心可取如 $(\rho, 0)$ 和 $(1, \rho)$ 。相邻半圆相切，所以这两个圆心距离为 $2\rho$ ： $(1 - \rho)^2 + \rho^2 = 4\rho^2$ 。这给出 $2\rho^2 + 2\rho - 1 = 0$ ，所以 $\rho = \tfrac{\sqrt3 - 1}{2}$ 。小圆半径为 $t$ ，圆心在 $\left(\tfrac12, \tfrac12\right)$ 。它与一个半圆相切时，到该半圆圆心的距离等于 $\rho + t$ 。这个距离是 $\sqrt{2 - \sqrt3}$ ，所以 $t = \sqrt{2 - \sqrt3} - \rho$ 。直径为 $2t = \sqrt6 - \sqrt2$ $- \sqrt3 + 1 = (\sqrt3 - 1)(\sqrt2 - 1)$ 。因此 $a + b + c + d = 3$ $+ (-1) + 2 + (-1) = 3$ ，正确答案是 A。

Let each semicircle have radius $\rho,$ with centers like $(\rho, 0)$ and $(1, \rho).$ Adjacent semicircles are tangent, so these centers are $2\rho$ apart: $(1 - \rho)^2 + \rho^2 = 4\rho^2.$ This gives $2\rho^2 + 2\rho - 1 = 0,$ so $\rho = \tfrac{\sqrt3 - 1}{2}.$ The small circle of radius $t$ sits at $\left(\tfrac12, \tfrac12\right),$ and it's tangent to a semicircle when its distance to that center equals $\rho + t.$ That distance is $\sqrt{2 - \sqrt3},$ so $t = \sqrt{2 - \sqrt3} - \rho,$ and the diameter is $2t = \sqrt6 - \sqrt2$ $- \sqrt3 + 1 = (\sqrt3 - 1)(\sqrt2 - 1).$ So $a + b + c + d = 3$ $+ (-1) + 2 + (-1) = 3.$ Therefore, the answer is A.

2025 AMC 10B 第 20 题

答案：A

难度评级：1930

解答：

其他年份的第 20 题