[Proposed, Niv 2] TST Mar 2015



  • Soient x,y x, y et z z des nombres réels strictement positifs. Montrer que :

    yxy+2y+1+zyz+2z+1+xzx+2x+134 \frac{y}{xy+2y+1} + \frac{z}{yz+2z+1} + \frac{x}{zx+2x+1} \leq \frac{3}{4}

    Étudier le cas d'égalité.



  • yxy+2y+1=\sum \frac{y}{xy+2y+1} = 1x+1+1y+1\sum \frac{1}{x+1+\frac{1}{y} +1} = 1x+1+y+1y14(1x+1+yy+1)=34\sum \frac{1}{x+1+\frac{y+1}{y} } \geq \frac{1}{4}(\sum \frac{1}{x+1}+ \sum \frac{y}{y+1})=\frac{3}{4}


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