A=\(\Sigma\dfrac{x}{\left(x+y+x+z\right)^2}\le\Sigma\dfrac{x}{4\left(x+y\right)\left(x+z\right)}\)=\(\dfrac{x\left(y+z\right)+y\left(x+z\right)+z\left(x+y\right)}{4\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)=\(\dfrac{2\left(xy+yz+zx\right)}{4\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)
=>A≤\(\dfrac{\left(xy+yz+zx\right)\left(x+y+z\right)}{6\left(x+y\right)\left(y+z\right)\left(z+x\right)}\)(x+y+z=3)
áp dụng bđt \(\dfrac{8}{9}\)(x+y+z)(xy+yz+zx)≤(x+y)(z+x)(z+y)
=>A≤\(\dfrac{\text{(x+y+z)(xy+yz+zx)}}{6.\dfrac{8}{9}\text{(x+y+z)(xy+yz+zx)}}\)=\(\dfrac{3}{16}\left(đpcm\right)\)
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