2024 AMC 12A Problem 20

Question

Points

P

and

Q

are chosen uniformly and independently at random on sides

\overline{AB}

and

\overline{AC},

respectively, of equilateral triangle

\triangle ABC.

Which of the following intervals contains the probability that the area of

\triangle APQ

is less than half the area of

\triangle ABC?

\left[\tfrac38,\tfrac12\right]

\left(\tfrac12,\tfrac23\right]

\left(\tfrac23,\tfrac34\right]

\left(\tfrac34,\tfrac78\right]

\left(\tfrac78,1\right]

Let

x=\dfrac{AP}{AB}

and

y=\dfrac{AQ}{AC}

be uniform on

[0,1];

the area ratio is

xy

Compute

P(xy\ge\tfrac12)=\displaystyle\int_{\frac{1}{2}}^{1}\left(1-\tfrac{1}{2x}\right)dx

and subtract from

1

With

x=\tfrac{AP}{AB}

and

y=\tfrac{AQ}{AC}

uniform on

[0,1],

the area ratio

\tfrac{[APQ]}{[ABC]}=xy.

The complementary event

xy\ge\tfrac12

requires

x\ge\tfrac12

and

y\in[\tfrac{1}{2x},1],

with probability

\begin{aligned} &\int_{\frac{1}{2}}^{1}\left(1-\frac{1}{2x}\right)dx \\ &=\frac12-\frac{\ln2}{2}\approx0.153. \end{aligned}

Therefore

P(xy\lt\tfrac12)\approx1-0.153=0.847,

which lies in

\left(\tfrac34,\tfrac78\right].

Thus, the correct answer is D.

Problem 20 in other years