MathDB

2

Part of 2015 European Mathematical Cup

Problems(2)

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

Source: European Mathematical Cup, 2015, Junior, P2

12/30/2016
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
inequalities
(a+b+c+3)/4 is bigger than sum of fractions

Source: European Mathematical Cup, 2015, Senior, P2

12/30/2016
Let a,b,ca,b,c be positive real numbers such that abc=1abc=1. Prove that a+b+c+34⩾1a+b+1b+c+1c+a.\frac{a+b+c+3}{4}\geqslant \frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}.
Dimitar Trenevski
inequalitiesalgebra