Prove inequality $(a_1+a_2+...+a_n)(\frac1{a_1}+\frac1{a_2}+...+\frac1{a_n})\geq n^2$ when $a_1,a_2,...,a_n$ are positive numbers.
2455
Prove that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$
66387
Prove $(x_1+.....+x_n)^2 \leq n(x_1^2 + .....+x_n^2)$
44233
How to prove this inequality $ x + \frac{1}{x} \geq 2 $
119668
How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$?
3030
Proving inequality $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}+\frac{3\sqrt[3]{abc}}{a+b+c} \geq 4$
12781
Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
76696
Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
14221
How can we prove it? ${a}{b}≤\frac{a^p}{p}+\frac{b^q}{q}$
111847
For $a,b,c>0$, prove that $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\ge a+b+c+\frac{4(a-b)^2}{a+b+c}$
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
137791
Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
19896
Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
104125
How prove this inequality??
89728
Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
58969
A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
26606
Prove the inequality $\,\frac{1}{\sqrt{1}+ \sqrt{3}} +\frac{1}{\sqrt{5}+ \sqrt{7} }+\ldots+\frac{1}{\sqrt{9997}+\sqrt{9999}}\gt 24$
120093
Proof of inequality $2(\sqrt{n+1}-\sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n} - \sqrt{n-1})$ using induction