Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Find the answer

Let $A=H_{1}, B=H_{6}+1$ . A real number $x$ is chosen randomly and uniformly in the interval $[A, B]$ . Find the probability that $x^{2}>x^{3}>x$ .

Your answer

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

Solution

$A=-1, B=3$ . For $x^{3}>x$ , either $x>1$ or $-1<x<0$ . However, for $x>1, x^{2}<x^{3}$ , so there are no solutions. $-1<x<0$ also satisfies $x^{2}>x^{3}$ , so our answer is $1 / 4$ .

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