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Existence of r

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September 9, 2010
inequality systemalgebrainequalitiesIMO Shortlist

Problem Statement

Suppose that x1,x2,…,xn{x_1, x_2, \dots , x_n} are positive integers for which x1+x2+⋯+xn=2(n+1)x_1 + x_2 + \cdots+ x_n = 2(n + 1). Show that there exists an integer rr with 0≤r≤n−10 \leq r \leq n - 1 for which the following n−1n - 1 inequalities hold: xr+1+⋯+xr+i≤2i+1,∀i,1≤i≤n−r;x_{r+1} + \cdots + x_{r+i} \leq 2i+ 1, \qquad \qquad \forall i, 1 \leq i \leq n - r; xr+1+⋯+xn+x1+⋯+xi≤2(n−r+i)+1,∀i,1≤i≤r−1.x_{r+1} + \cdots + x_n + x_1 + \cdots+ x_i \leq 2(n - r + i) + 1, \qquad \qquad \forall i, 1 \leq i \leq r - 1. Prove that if all the inequalities are strict, then rr is unique and that otherwise there are exactly two such r.r.
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