Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq\sqrt{x}+\sqrt{y}$
14952
Prove: $\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{x}}\geq \sqrt{x}+\sqrt{y}$
105641
Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
83531
Prove $\iint_{x^{2}+y^{2} \leq 1} e^{x^{2}+y^{2}} d x d y \leq\left[\int_{-\frac{\sqrt{\pi}}{2}}^{\frac{\sqrt{\pi}}{2}} e^{x^{2}} d x\right]^{2} $
87111
Prove $\sqrt{2}$ is between $\dfrac{a}{b}$ and $\dfrac{a+2b}{a+b}$
49750
Proof of the square root inequality $2\sqrt{n+1}-2\sqrt{n}<\frac{1}{\sqrt{n}}<2\sqrt{n}-2\sqrt{n-1}$
66247
Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$
77130
How to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|<2\sqrt{|x-y|}$?
103851
Show that $\frac{xy}{z} + \frac{xz}{y} + \frac{yz}{x} \geq x+y+z $ by considering homogeneity
26876
How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$?
44705
Prove that $\frac{1}{a^3(b+c)}+\frac{1}{b^3(a+c)}+\frac{1}{c^3(a+b)}\ge \frac32$
152053
Proof that $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a+b+c$
100995
For what $x$ : $2\sqrt{x}+\dfrac{1}{\sqrt{x+1}} \leq 2\sqrt{x+1}$
18016
Proof of the following inequality $ \frac{x - y}{\log x - \log y} > \sqrt{xy} $, $x>y$.
146546
Proving for all integer $n \ge 2$, $\sqrt n < \frac{1}{\sqrt 1} + \frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+\cdots+\frac{1}{\sqrt n}$
19896
Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
137791
Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
94119
Prove that $xy \leq\frac{x^p}{p} + \frac{y^q}{q}$