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It's possible to find v_i (IMO SL 1987-P7)

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August 19, 2010
algebrainequality systemIMO Shortlist

Problem Statement

Given five real numbers u0,u1,u2,u3,u4u_0, u_1, u_2, u_3, u_4, prove that it is always possible to find five real numbers v0,v1,v2,v3,v4v0, v_1, v_2, v_3, v_4 that satisfy the following conditions:
(i)(i) u_i-v_i \in \mathbb N,   0 \leq i \leq 4
(ii)(ii) ∑0≤i<j≤4(vi−vj)2<4.\sum_{0 \leq i<j \leq 4} (v_i - v_j)^2 < 4.
Proposed by Netherlands.
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