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Minimise x^2+y^2+z^2+ mxy + nxz + pyz

Source: European Mathematical Cup, 2015, Junior, P2

December 30, 2016
inequalities

Problem Statement

Let m,n,pm, n, p be fixed positive real numbers which satisfy mnp=8mnp = 8. Depending on these constants, find the minimum of x2+y2+z2+mxy+nxz+pyz,x^2+y^2+z^2+ mxy + nxz + pyz, where x,y,zx, y, z are arbitrary positive real numbers satisfying xyz=8xyz = 8. When is the equality attained? Solve the problem for: [*]m=n=p=2,m = n = p = 2, [*] arbitrary (but fixed) positive real numbers m,n,p.m, n, p.
Stijn Cambie
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