Inequality

[man111]

If [tex]x_{i}\in \left[0,1\right]\;\forall i=1,2,3,...,n[/tex] and [tex]x_{1}+x_{2}+x_{3}+....+x_{n} = 1,[/tex] Then prove that

[tex]\displaystyle \frac{\sqrt{1-x_{1}}+\sqrt{1-x_{2}}+......+\sqrt{1-x_{n}}}{\sqrt{x_{1}}+\sqrt{x_{2}}+.....+\sqrt{x_{n}}}\geq \sqrt{n-1}[/tex]

[ptj]

[tex]\sqrt{\frac{(\sqrt{x_{1}})^2+(\sqrt{x_{2}})^2+.....+(\sqrt{x_{n}})^2}{n}} \ge \frac{\sqrt{x_{1}}+\sqrt{x_{2}}+.....+\sqrt{x_{n}}}{n}[/tex]

[tex]\sqrt{\frac{x_{1}+x_{2}+.....+x_{n}}{n}}\ge \frac{\sqrt{x_{1}}+\sqrt{x_{2}}+.....+\sqrt{x_{n}}}{n}[/tex]

[tex]\sqrt{\frac{1}{n}}\ge \frac{\sqrt{x_{1}}+\sqrt{x_{2}}+.....+\sqrt{x_{n}}}{n}[/tex]

[tex]\frac{1}{\sqrt{x_{1}}+\sqrt{x_{2}}+.....+\sqrt{x_{n}}}\ge\sqrt{\frac{1}{n}}[/tex]
...

[Knowledge Greedy]

А какво се крие в многоточието?

[nevrodermit]

[tex]\sqrt{1-x_1}=\sqrt{x_2+x_3+\cdots +x_n}[/tex]
.............
[tex]\sqrt{1-x_n}=\sqrt{x_1+x_2+\cdots +x_{n-1}}[/tex]

We have that
[tex]\sqrt{1-x_1} -\sqrt{n-1}\sqrt{x_1} = \sqrt{x_2+x_3+\cdots +x_n} -\sqrt{n-1}\sqrt{x_1}=\frac{x_2+x_3+\cdots + x_n -(n-1)x_1}{\sqrt{x_2+x_3+\cdots +x_n} +\sqrt{n-1}\sqrt{x_1}}=\frac{x_2+x_3+\cdots + x_n +x_1-nx_1}{\sqrt{x_2+x_3+\cdots +x_n} +\sqrt{n-1}\sqrt{x_1}}=\frac{1-nx_1}{\sqrt{x_2+x_3+\cdots +x_n} +\sqrt{n-1}\sqrt{x_1}}[/tex]
Also we have
[tex]\sqrt{x_2+x_3+\cdots +x_n} +\sqrt{n-1}\sqrt{x_1} \le \sqrt{2}.\sqrt{1+(n-2)x_1} \le \sqrt{2}.\sqrt{(n-2)+(n-2)x_1}=\sqrt{2(n-2)}\sqrt{1+x_1}\le \sqrt{2(n-2)}\sqrt{2}=2\sqrt{n-2}[/tex]

Wring the same for all [tex]i[/tex] we get
[tex]\sqrt{1-x_1} -\sqrt{n-1}\sqrt{x_1} \ge \frac{1-nx_1}{2\sqrt{n-2}}[/tex]
[tex]\sqrt{1-x_2} -\sqrt{n-1}\sqrt{x_2} \ge \frac{1-nx_2}{2\sqrt{n-2}}[/tex]
.........
[tex]\sqrt{1-x_n} -\sqrt{n-1}\sqrt{x_1} \ge \frac{1-nx_n}{2\sqrt{n-2}}[/tex]

Summing them we get
[tex]\sum_{i}\sqrt{1-x_i} - \sum_{i}\sqrt{n-1}\sqrt{x_i} \ge \frac{\sum_{i}1-n\sum_{i}x_1}{2\sqrt{n-2}} \Rightarrow[/tex]
[tex]\sum_{i}\sqrt{1-x_i} - \sum_{i}\sqrt{n-1}\sqrt{x_i} \ge \frac{n-n}{2\sqrt{n-2}} \Rightarrow[/tex]
[tex]\sum_{i}\sqrt{1-x_i} \ge \sum_{i}\sqrt{n-1}\sqrt{x_i} \Rightarrow[/tex]
[tex]\sum_{i}\sqrt{1-x_i} \ge \sqrt{n-1}\sum_{i}\sqrt{x_i} \Rightarrow[/tex]
[tex]\frac{\sum_{i}\sqrt{1-x_i}}{\sum_{i}\sqrt{x_i}} \ge \sqrt{n-1}[/tex]

P.S. I tried maybe 10 different ways to solve this and in the end I pieced up this

[man111]

Very Nice explanation nevrodermit.