Question

[Phạm Dương Ngọc Nhi]

Cho x, y, z là số dương thỏa mãn xyz = 1. Chứng minh rằng :

\(\left(\frac{1+x}{2}\right)^n+\left(\frac{1+y}{2}\right)^n+\left(\frac{1+z}{2}\right)^{^n}\ge3\)

Answer 1

[Nguyễn Việt Lâm]

\(VT=\left(\frac{1+x}{2}\right)^n+\left(\frac{1+y}{2}\right)^n+\left(\frac{1+z}{2}\right)^n\)

\(VT\ge\left(\frac{2\sqrt{x}}{2}\right)^n+\left(\frac{2\sqrt{y}}{2}\right)^n+\left(\frac{2\sqrt{z}}{2}\right)^n\)

\(VT\ge x^{\frac{n}{2}}+y^{\frac{n}{2}}+z^{\frac{n}{2}}\ge3\sqrt[3]{\left(xyz\right)^{\frac{n}{2}}}=3\)

Dấu "=" xảy ra khi \(x=y=z=1\)