This is the problem from Vojtěch Jarník Competition 2006. Given real numbers $0=x_1,x_2<\dots
$$x_{i+1}-x_{i}\leq h$$ $$x_{i}-x_{i-1}\leq h$$ $$\dots$$ $$\frac{x_{i+1}-x_{i-m}}{m+1}\leq h$$
$$x_{2i-1}>\frac{x_{2i-1}+x_{2i-2}}{2}>x_{2i-1}x_{2i-2}>x_{2i-2}$$ $$x_{2i}>\frac{x_{2i}+x_{2i-1}}{2}>x_{2i}x_{2i-1}>x_{2i-1}$$ $$x_{2i-1}-x_{2i}>\frac{x_{2i-2}-x_{2i}}{2}>x_{2i-1}(x_{2i-2}-x_{2i})>x_{2i-2}-x_{2i-1}\geq -h$$
$$\frac{1+h}{2}\geq \frac{x_i-x_{i-n}+n}{2n}=\dfrac{\dfrac{x_i-x_{i-n}}{n}+1}{2}>\frac{x_i-x_{i-n}}{n}$$ I would not mind to see not only hints but full proofs as well, in case I am nowhere near the truth