Selected Problems from the Final Round of National Olympiad

For simplicity, denote $A=x_1{x}_2+x_1{x}_3+x_1{x}_4+x_2{x}_3+x_2{x}_4+x_3{x}_4$ and $B=x_1{x}_2+x_2{x}_3+x_3{x}_4+x_4{x}_1$.

Choose $x_1=x_3=u$ and $x_2=x_4=1$. Then $A\ge x_1{x}_3=u^2$ and $B=4u$. Hence $\frac{A}{B}\ge \frac{u}{4}$. As $u$ can be arbitrarily large, no constant $C$ such that $A\le CB$ exists.