Proving that $\frac{1}{a_1....a_n} \geq (n-1)^n $ under certain conditions
12781
Prove the following inequality $\sqrt[n]{x_1x_2...x_n}\ge (n-1)$
34081
Prove that $|f''(\xi)|\geqslant\frac{4|f(a)-f(b)|}{(b-a)^2}$
68705
Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$
10902
Showing that $ \prod\limits_{i=0}^{n} {n \choose i}\leq(\frac{2^n-2}{n-1})^{n-1} $
138819
Prove that $\phi(n) \geq \sqrt{n}/2$
80826
To prove $\left|\frac{p_n(z)}{q_m(z)}\right|\leq \frac{M}{|z|^{m-n}}$ for some $M>0$
61670
Proving that $(b_n) \to b$ implies $\left(\frac{1}{b_n}\right) \to \frac{1}{b}$
44233
How to prove this inequality $ x + \frac{1}{x} \geq 2 $
3110
Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$
149056
Prove that for all $n\geqslant 1$ we have $F_n<{\left(\frac 74\right)}^n$.
65249
An interesting exercise about converging positive series, involving $\sum_{n\geq 1}a_n^{\frac{n-1}{n}}$
116387
How to prove that $\Gamma(x) \leq x^x$ for $x \geq 1$
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
78144
How to prove the inequality $a_1^{a_1}a_2^{a_2}\cdots a_n^{a_n} > \left(\frac{a_1+\dots+a_n}{n}\right)^{a_1+\dots+a_n}$?
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
25648
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
108298
Prove $\sum_{i=1}^{i=n}\frac{a_i}{S-a_i}\geqslant \frac{n}{n-1}$ and $\sum_{i=1}^{i=n}\frac{S-a_i}{a_i}\geqslant n(n-1)$.