2017 AIME I 第 6 题

Question

一个圆外接于一个等腰三角形，该三角形两个相等角的度数为

x

。在圆上独立且均匀随机选取两个点，并连接它们作一条弦。该弦与三角形相交的概率为

\frac{14}{25}

。求

x

的最大可能值与最小可能值之差。

A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure

x.

Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is

\frac{14}{25}.

Find the difference between the largest and smallest possible values of

x.

答案

答案：48

知识点：几何概率圆周角二次方程

难度评级：2650

小提示：

三个顶点把圆分成度数为

2x

、

2x

和

360 - 4x

的弧；弦恰好在两个点落在同一段弧内时避开三角形

The vertices cut the circle into arcs of

2x,

2x,

and

360 - 4x

degrees, and the chord misses the triangle exactly when both points land in the same arc

大提示：

令

y = \frac{x}{180}

，不相交的概率给出

2y^2 + (1 - 2y)^2 = \frac{11}{25}

，这是一个有两个有效根的二次方程

With

y = \frac{x}{180},

the miss probability gives

2y^2 + (1 - 2y)^2 = \frac{11}{25},

a quadratic with two valid roots

解答：

三角形的每个圆周角截得一段度数为其两倍的弧，所以三个顶点把圆分成度数为

2x

、

2x

和

360 - 4x

的弧。弦不与三角形相交，当且仅当两个随机点落在同一段弧内，其概率为

\begin{aligned} &\left(\frac{2x}{360}\right)^2 + \left(\frac{2x}{360}\right)^2 \\ &\quad {}+ \left(\frac{360 - 4x}{360}\right)^2 \\ &= 1 - \frac{14}{25} = \frac{11}{25} \end{aligned}\text{。}

令

y = \frac{x}{180}

，得到

2y^2 + (1 - 2y)^2 = \frac{11}{25}

，化简为

75y^2 - 50y + 7 = 0

，根为

y = \frac{1}{5}

和

y = \frac{7}{15}

。它们给出

x = 36

和

x = 84

，两者都可以作为等腰三角形的底角。

所求差为

84 - 36 = 48

。

Each inscribed angle of the triangle subtends an arc of twice its measure, so the vertices split the circle into arcs of

2x,

2x,

and

360 - 4x

degrees. The chord fails to intersect the triangle exactly when both random points fall in the same arc, which has probability

\begin{aligned} &\left(\frac{2x}{360}\right)^2 + \left(\frac{2x}{360}\right)^2 \\ &\quad {}+ \left(\frac{360 - 4x}{360}\right)^2 \\ &= 1 - \frac{14}{25} = \frac{11}{25}. \end{aligned}

Setting

y = \frac{x}{180},

this reads

2y^2 + (1 - 2y)^2 = \frac{11}{25},

which simplifies to

75y^2 - 50y + 7 = 0,

with roots

y = \frac{1}{5}

and

y = \frac{7}{15}.

These give

x = 36

and

x = 84,

both legitimate base angles of an isosceles triangle.

The requested difference is

84 - 36 = 48.

2017 AIME I 第 6 题

答案：48

难度评级：2650

小提示：

大提示：

解答：

其他年份的第 6 题