62nd Ukrainian National Mathematical Olympiad, Third Round, Second Tour

Consider numbers $x_1=\sqrt{2},x_2=2\sqrt{2},...,x_{n+1}=(n+1)\sqrt{2}$. Suppose that for each $k=1,n+1$ at least one of the numbers $x_k+a_1,x_k+a_2,...,x_k+a_n$ is rational. As we have $n$ numbers, and try $n+1$ options, from the Dirichlet principle some two numbers of form $x_i+a_i$ and $x_j+a_j$ will be rational. But then their difference also will be rational, so the number $$(x_j+a_i)-(x_i+a_j)=x_j-x_i=(j-i)\sqrt{2}$$ will be rational, which isn't true. This contradiction finishes the proof, as for some $x_k=k\sqrt{2}$ all integers $x_k+a_1,x_k+a_2,...,x_k+a_n$ are irrational.