CIIM 2013 Problem 4

Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real number and $F,G:(0,\infty)\to(0,\infty)$ be to differentiable and positive functions that satisfy the identities: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$ $$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$ Prove that if $0 < x_1 \leq x_2$ and $0 < y_2 \leq y_1$ , then $F(x_1,x_2) \leq F(x_2,y_2)$ and $G(x_1,y_1) \geq G(x_2,y_2).$