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2006 China Second Round Olympiad Test 1 #14

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September 28, 2014

Problem Statement

Let 20062006 be expressed as the sum of five positive integers x1,x2,x3,x4,x5x_1, x_2, x_3, x_4, x_5, and S=∑1≤i<j≤5xixjS=\sum_{1\le i<j\le 5}x_ix_j. <spanclass=′latex−bold′>(A)</span> <span class='latex-bold'>(A)</span> What value of x1,x2,x3,x4,x5x_1, x_2, x_3, x_4, x_5 maximizes SS? <spanclass=′latex−bold′>(A)</span> <span class='latex-bold'>(A)</span> Find, with proof, the value of x1,x2,x3,x4,x5x_1, x_2, x_3, x_4, x_5 which minimizes of SS if ∣xi−xj∣≤2|x_i-x_j|\le 2 for any 1≤i1\le i, j≤5j\le 5.
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