Question

Cho a,b,c là các số thực dương thỏa mãn a+b+c=3 CMR

(a²+3)/(b+c)+(b²+3)/(c+a)+(c²+3)/(a+b)≥6

Answer 1

Áp dụng liên tiếp bất đẳng thức Cauchy-Schwarz ta có:

\(\dfrac{a^2+3}{b+c}+\dfrac{b^2+3}{c+a}+\dfrac{c^2+3}{a+b}\)

\(=\dfrac{a^2}{b+c}+\dfrac{3}{b+c}+\dfrac{b^2}{c+a}+\dfrac{3}{c+a}+\dfrac{c^2}{a+b}+\dfrac{3}{a+b}\)

\(=\left(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\right)+3\left(\dfrac{1}{b+c}+\dfrac{1}{c+a}+\dfrac{1}{a+b}\right)\)

\(\ge\dfrac{\left(a+b+c\right)^2}{2\left(a+b+c\right)}+3.\dfrac{\left(1+1+1\right)^2}{2\left(a+b+c\right)}\)

\(=\dfrac{a+b+c}{2}+\dfrac{27}{2\left(a+b+c\right)}=\dfrac{3}{2}+\dfrac{27}{6}=6\)