A function g: is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers m and n. Let f be an adjective function such that the value of $f(1)+f(2)+f(3)+...+f(30)$ is minimized. Find the smallest possible value of $f(25)$.
My approach: - $\\ f(1)+f(2)+f(3)+...+f(30) \\ = (f(1)+f(30))+(f(2)+f(29))+...+(f(15)+f(16)) \\ \geq 30^2+1+29^2+1+...16^2+1$
If minimized, $f(1)+f(2)+f(3)+...+f(30)= \Sigma_{i=16}^{30}(i^2+1)$
Now, if $j\in {1,2,..,15}$ and $i\in {1,2,..,15}, f(j)+f(i) \geq i^2+1$
After that, how can I approach the problem?