This one, by Oleg Golberg, appeared in the 6^{th} issue of Mathematical Reflections (MR) 2007. I don’t have a solution yet, but I think I should be able to solve it sooner or later. If you find a solution, you should send it to MR by Jan 19. Here is the problem anyway.

For all integers k, n \geq 2, prove that \\ \displaystyle \sqrt[n]{1 + \frac{n}{k}} \leq \frac1{n} \log \big( 1+\frac{n}{k-1} \big) + 1.