45th Mongolian Mathematical Olympiad

$${\left(\frac{1}{x^2(1-x)+y^2(1-y)+z^2(1-z)}\right)}^2\le$$

$$\le \frac{1}{x(1-x{)}^2}+\frac{1}{y(1-y{)}^2}+\frac{1}{z(1-z{)}^2}.$$ Let's consider $f(x)=\frac{1}{x^2}$ function. Because $f^{\prime \prime }(x)=\frac{6}{x^4}>0$, hence apply Jensen's inequality and we choose $\alpha =x$, $\beta =y$, $\gamma =z$, $x_1=x(1-x)$, $x_2=y(1-y)$, $x_3=z(1-z)$: $$f(\alpha x_1+\beta x_2+\gamma x_3)\le \alpha f(x_1)+\beta f(x_2)+\gamma f(x_3).$$ This is the desired result.