Problem

Source: Uzbekistan NO 2013, P2

Tags: trigonometry, inequalities proposed, algebra, Inequality



Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.