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Se dau numerele reale \( x,y,z \geq 0 \). Sa se demonstreze inegalitatea
\( \frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} \geq \sqrt{2} \cdot \sqrt{2-\frac{7xyz}{(x+y)(y+z)(z+x)}} \)

Andrei Ciupan, Shortlist ONM 2008

Ridicand la patrat si efectind calculele obtinem:

\( \sum_{cyc} {(x^6+2x^5y+2x^5z-x^4y^2-x^4z^2+x^4yz-4x^3y^3)}\geq0 \) sau

\( \sum_{cyc}{(x^6-x^5y-x^5z+x^4yz)}+\sum_{sym}{(x^5y-x^4y^2)}+\sum_{sym}{(x^5y-x^3y^3)}\geq0 \)

si care este adevarata folosind Schur si Muirhead.