1. Find the smallest value of $x$ such that $a \geq 14 \sqrt{a}-x$ for all nonnegative $a$.
Solution:
49

We want to find the smallest value of $x$ such that $x \geq 14$ sqrta $-a$ for all $a$. This is just the maximum possible value of $14 \sqrt{a}-a=49-(\sqrt{a}-7)^{2}$, which is clearly 49 , achieved when $a=49$.