Question

Cho a,b,c là các số thực dương thỏa mãn:a+b+c=1.CMR:\(\frac{a^7+b^7}{a^5+b^5}+\frac{b^7+c^7}{b^5+c^5}+\frac{c^7+a^7}{c^5+a^5}\)≥\(\frac{1}{3}\)

Answer 1

Ta có đánh giá: \(\frac{a^7+b^7}{a^5+b^5}\ge\frac{a^2+b^2}{2}\)

\(\Leftrightarrow2a^7+2b^7\ge a^7+b^7+a^5b^2+a^2b^5\)

\(\Leftrightarrow a^5\left(a^2-b^2\right)-b^5\left(a^2-b^2\right)\ge0\)

\(\Leftrightarrow\left(a-b\right)^2\left(a+b\right)\left(a^4+a^3b+a^2b^2+ab^3+b^4\right)\ge0\) (luôn đúng)

Tương tự \(\frac{b^7+c^7}{b^5+c^5}\ge\frac{b^2+c^2}{2}\) ; \(\frac{c^7+a^7}{c^5+a^5}\ge\frac{a^2+c^2}{2}\)

\(\Rightarrow VT\ge a^2+b^2+c^2\ge\frac{\left(a+b+c\right)^2}{3}=\frac{1}{3}\)

Dấu "=" xảy ra khi \(a=b=c=\frac{1}{3}\)