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min (a-b)^2 + 2(a-c)^2+ 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2

Source: 2018 Romanian NMO grade VIII P2

September 4, 2024
algebrainequalitiesnumber theory

Problem Statement

Let a,b,c,da, b, c, d be natural numbers such that a+b+c+d=2018a + b + c + d = 2018. Find the minimum value of the expression: E=(a−b)2+2(a−c)2+3(a−d)2+4(b−c)2+5(b−d)2+6(c−d)2.E = (a-b)^2 + 2(a-c)^2 + 3(a-d)^2+4(b-c)^2 + 5(b-d)^2 + 6(c-d)^2.
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