Prove for that $(1 + \frac{x}{p})^p < (1 + \frac{x}{q})^q$ [without Bernoulli inequality or integrals]
121548
proof of an inequality $ \left( 1+ \frac{x}{p} \right) ^{p}$
349
Prove $| \sum_{i \leq n} \frac{\mu(i)}{i} | \leq 1$
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
36837
Prove $\frac{1+a^{2}}{1+b+c^{2}} +\frac{1+b^{2}}{1+c+a^{2}} +\frac{1+c^{2}}{1+a+b^{2}} \geq 2$
98454
Prove that, $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\ge 4$ where we do not use AM-GM inequality on the given statement to prove it.
155341
Proving or disproving inequality $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y} \ge x + y + z $
115335
How can we prove this integral inequality?
3109
Prove that $(p+q)^m \leq p^m+q^m$
154716
Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
13902
Show that $\frac{1}{x_1^2}+\frac{1}{4\cdot x_2^2}+\ldots+\frac{1}{n^2\cdot x_n^2}\leq\frac{3n-2}{2n-1}$.
107494
prove $\frac {ab}{a+b} \geq \sum ^n_{i=1} \frac{a_ib_i}{a_i+b_i}$
118884
The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$
56315
Prove $\frac{|a+b|}{1+|a+b|}<\frac{|a|}{1+|a|}+\frac{|b|}{1+|b|}$.
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
2491
For $0 < x < \pi/2 $, prove that $x + \frac{x^{3}}{3} <\tan x $
40420
Proof by induction of Bernoulli's inequality: $(1 + x)^n \geq 1 + nx$
45742
Prove that ${\left(\frac{x+1}{2}\right)}^{x+1} \leq x^x$