ILL 1987 Problem 46

Given five real numbers $u_0, u_1, u_2, u_3, u_4$, prove that it is always possible to find five real numbers $v0, v_1, v_2, v_3, v_4$ that satisfy the following conditions:

$(i)$ $u_i-v_i \in \mathbb N, 0 \leq i \leq 4$

$(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$

Proposed by Netherlands.