This problem comes from question 5 in the PUMAC Algebra A competition (link here):
Suppose $w, x, y, z$ satisfy $$w+x+y+z=25$$ $$wx+wy+wz+xy+xz+yz=2y+2x+193$$ The largest possible value of $w$ can be expressed in lowest terms as $w_1/w_2$ for some integers $w_1, w_2 > 0$. Find $w_1+w_2$.
How would you find the solutions to this equation? I don't realy understand how to solve the problem, and their solution confused me a lot. I understand that you would have to use inequalities for $w$ in order to find it's largest possible value, but I don't know how to find that inequality.
Any help/hints is greatly appreciated.