Question

Cho a,b,c,d > 0 và abcd=1.CMR: a^2 + b^2 + c^2 + d^2 + a(b+c) + b(c+d) + d(c+a) >= 10

Answer 1

Áp dụng bđt Cô-si: \(a^2+b^2+c^2+d^2\)\(\ge4\sqrt[4]{a^2.b^2.c^2.d^2}\)\(=4\sqrt[4]{\left(abcd\right)^2}=4\sqrt[4]{1^2}=4;\)

\(a\left(b+c\right)+b\left(c+d\right)+d\left(c+a\right)=ab+ac+bc+bd+dc+da\)

\(\ge6\sqrt[6]{ab.ac.bc.bd.dc.da}=6\sqrt[6]{\left(abcd\right)^3}=6\sqrt[6]{1^3}=6\)

=>\(a^2+b^2+c^2+d^2\)\(a\left(b+c\right)+b\left(c+d\right)+d\left(c+a\right)\ge4+6=10\)

Dấu "=" xảy ra khi a=b=c=d=1