[Toán 10] Bài khó

A

angleofdarkness

Có $(x-y)^2+(x-z)^2+(x-t)^2+(y-z)^2+(y-t)^2+(z-t)^2$ \geq 0.

\Leftrightarrow $3(x^2+y^2+z^2+t^2)-2(xy+xz+zt+yz+yt+zt)$ \geq 0.

\Leftrightarrow $3(x^2+y^2+z^2+t^2)$ \geq $2(xy+xz+zt+yz+yt+zt).$

\Leftrightarrow $x^2+y^2+z^2+t^2+2(xy+xz+zt+yz+yt+zt).$ \geq $\dfrac{2}{3}(xy+xz+zt+yz+yt+zt)+2(xy+xz+zt+yz+yt+zt.$

\Leftrightarrow $(x+y+z+t)^2$ \geq $\dfrac{8}{3}.(xy+xz+zt+yz+yt+zt).$
 
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