Problem 599 (difficulty: 7/10)

Prove that for the reals \(\displaystyle 0<a<b\) the equality \(\displaystyle a^b=b^a\) holds if and only if there is a positive number \(\displaystyle x\) for which \(\displaystyle a=\left(1+\frac1x\right)^x\) and \(\displaystyle b=\left(1+\frac1x\right)^{x+1}\).

Give me another random problem!