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Integer solutions of the system of inequalities

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September 20, 2010
combinatoricsalgebracountinginequality systemIMO Shortlist

Problem Statement

Let nn be a positive integer. How many integer solutions (i,j,k,l), 1≤i,j,k,l≤n(i, j, k, l) , \ 1 \leq i, j, k, l \leq n, does the following system of inequalities have:
1≤−j+k+l≤n1 \leq -j + k + l \leq n1≤i−k+l≤n1 \leq i - k + l \leq n1≤i−j+l≤n1 \leq i - j + l \leq n1≤i+j−k≤n ?1 \leq i + j - k \leq n \ ?
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