Question

Cho a,b,c,d,e dương.CMR \(\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+e}+\frac{d}{e+a}+\frac{e}{a+b}>=\frac{5}{2}\)

Answer 1

đặt vt=A

\(A>=\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}\right)^2}{2\left(a+b+c+d+e\right)}\)(bdt cauchy schwarz)

=>\(\frac{2A}{5}>=\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{e}+\sqrt{d}\right)^2}{5\left(a+b+c+d+e\right)}>\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}\right)^2}{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}+\sqrt{d}+\sqrt{e}\right)^2}=1\)(gợi ý:chỗ này dựa vào bdt bunhiacopxki)

=>\(A>=\frac{5}{2}\)