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Harvard-MIT Mathematics Tournament

Algebra Difficulty 4.2 AIME Find the answer United States

Problem:

Let $p$ be the answer to this question. If a point is chosen uniformly at random from the square bounded by $x=0$ , $x=1$ , $y=0$ , and $y=1$ , what is the probability that at least one of its coordinates is greater than $p$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The probability that a randomly chosen point has both coordinates less than $p$ is $p^{2}$ , so the probability that at least one of its coordinates is greater than $p$ is $1 - p^{2}$ . Since $p$ is the answer to this question, we have $1 - p^{2} = p$ , and the only solution of $p$ in the interval $[0,1]$ is $\frac{\sqrt{5} - 1}{2}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.