This problem is from Antonio Caminha's book "Números reais" (real numbers), a Brazillian book for contest math preparation.
Problem:
If $a$,$b$,$c$ and $d$ are non-negative real numbers such that $a \le 1$, $a + b \le 5$, $a + b + c\le 14$ and $a + b + c +d\le 30$, prove that
$$\sqrt{a}+ \sqrt{b} + \sqrt{c} + \sqrt{d} \le 10 $$
In the problem, it suggests to use the Abel's inequality. But I couldn't find out the solution, I just noticed that if we take $a=1$, $b=4$, $c=9$ and $d=16$, we have $\sqrt{a}+ \sqrt{b} + \sqrt{c} + \sqrt{d} = 10 $.
The problem is already solved here: Inequality question. However, there the problem is solved using advanced math (like polyhedral domain, stationary points, etc). Apparantly, this problem can be solved just with elementary math (things at the level of Abel's inequality, AM–GM inequality, etc), since the book is at high school level or even lower.