Question

[Nhok_baobinh]

CMR: \(\left(a+b+c+d\right)^2\ge\frac{8}{3}\left(ab+ac+ad+bc+bd+cd\right)\\ \)

Answer 1

[Phúc]

(a+b+c+d)2\(\ge\frac{8}{3}\)(ab+ac+ad+bc+bd+cd)

<=>(a+b)²+2(a+b)(c+d)+(c+d)²\(\ge\).....

<=>a²+b²+c²+d²+2(ab+ac+ad+bc+bd+cd)\(\ge\)....

<=>3a²+3b²+3c²+3d²+6(ab+ac+ad+bc+bd+cd)\(\ge\)8(ab+ac+ad+bc+bd+cd)

<=> 3a²+3b²+3c²+3d²-2ab -2ac-2bc-2ad-2bd-2cd\(\ge\)0

<=> (a²-2ab+b²)+(a²-ac+c²)+(a²-2ad+d²)+(b²-2bc+c²)+(b²-2bd+d²)+(c²-2cd+d²)>=0

<=> (a-b)²+(a-c)²+(a-d)²+(b-c)²+(b-d)²+(c-d)²>=0 (DPCM)

Dau ''='' xay ra khi a=b=c=d