BMO 2017

The hunter can catch the grasshopper. Here is the strategy for him. Let $f$ be any bijection between the set of positive integers and the set $\{((x,y),(u,v)):x,y,u,v\in \mathbb{Z}\}$, and denote $$f(t)=((x_t,y_t),(u_t,v_t))$$ In the second $t$, the hunter should hunt at the point $(x_t+t{u}_t,y_t+t{v}_t)$. Let us show that this strategy indeed works.

Assume that the grasshopper starts at the point $(x^{\prime },y^{\prime })$ and that the jump vector is $(u^{\prime },v^{\prime })$. Then in the second $t$ the grasshopper is at the point $(x^{\prime }+t{u}^{\prime },y^{\prime }+t{v}^{\prime })$. Let $$t^{\prime }=f^{-1}((x^{\prime },y^{\prime }),(u^{\prime },v^{\prime }))$$ The hunter's strategy dictates that in the second $t^{\prime }$ he searches for the grasshopper at the point $(x_{t^{\prime }}+t^{\prime }{u}_{t^{\prime }},y_{t^{\prime }}+t^{\prime }{v}_{t^{\prime }})$, which is actually $(x^{\prime }+t^{\prime }{u}^{\prime },y^{\prime }+t^{\prime }{v}^{\prime })$, and this is precisely the point where the grasshopper is in the second $t^{\prime }$. This completes the proof.