Before solving the problem, I note the following:
Lemma: If n non-negative terms sum up to k, then the sum of their square roots is maximized if each term = k/n. The closer the individual terms are to the average, the larger the sum of the square. I could prove this with calculus, but will instead just use and example. Let n = 2 and k = 5.
sqrt(0) + sqrt(5) = 2.23
sqrt(1) + sqrt(4) = 3.00
sqrt(2) + sqrt(3) = 3.15
sqrt(2.5) + sqrt(2.5) = 3.16
So, what does this have to do with our problem?
Well, consider the sum of the square roots of a,b,c and d where a+b+c+d<=30. Clearly, the sum of the square roots is maximized if a+b+c+d=30. But a+b+c<= 14. The sum of the squares is therefore maximized if d = 30 - 14 = 16 and a+b+c = 14. It is for instance possible for d = 17 and a+b+c = 13, but this is necessarily a smaller sum of square roots (using our lemma).
Furthermore, using the same logic, if a+b+c = 14 and a+b<=9, then the sum of the square roots of a,b,c is maximized if c = 14 - 5 = 9 and a+b = 5
Furthermore, using the same logic, if a+b = 5 and a<=1, then the sum of the square roots of a,b is maximized if b = 5 - 1 = 4 and a = 1
So, the sum of the square roots is maximized if a = 1, b = 4, c = 9, and d = 16. And the sum of those square roots is 1+2+3+4 = 10.
So, finally, for any other values of a,b,c,d the sum of their square roots is necessarily less than or equal to the maximum sum of the square roots, which is 10.
With just a little hand-waving (spoiler)
