IMO Shortlist 1987 problem 7
Dodao/la:
arhiva2. travnja 2012. Given five real numbers
![u_0, u_1, u_2, u_3, u_4](/media/m/2/0/a/20abe1b6ac4a4d908b4e87b3f5b668f0.png)
, prove that it is always possible to find five real numbers
![v0, v_1, v_2, v_3, v_4](/media/m/9/a/6/9a66a7a94e96b46def30fcae4fa32081.png)
that satisfy the following conditions:
![\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.](/media/m/f/6/5/f65fa63f271685b264eb13f26a24c42c.png)
Proposed by Netherlands.
%V0
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:
$(i)$ $u_i-v_i \in \mathbb N, \quad 0 \leq i \leq 4$
$(ii)$ $\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.$
Proposed by Netherlands.
Izvor: Međunarodna matematička olimpijada, shortlist 1987