Prove that $(a+b\sqrt[3]{2}+c\sqrt[3]{4})^{-1}$ with a,b,c∈Q is a number of the form $d+e\sqrt[3]{2}+f\sqrt[3]{4}$ with $d,e,f∈Q$
124564
Prove the ring $a+b\sqrt[3]{2}+c\sqrt[3]{4}$ has inverse and is a field
75639
Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
15116
Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$
118507
Prove $\frac{ab}{1+c^2}+\frac{bc}{1+a^2}+\frac{ca}{1+b^2}\le\frac{3}{4}$ if $a^2+b^2+c^2=1$
92300
For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
117046
Show that $\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}}+\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}}+\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}$
25648
Prove $(a_1+b_1)^{1/n}\cdots(a_n+b_n)^{1/n}\ge \left(a_1\cdots a_n\right)^{1/n}+\left(b_1\cdots b_n\right)^{1/n}$
97057
Show $a^2+b^2+c^2=1$ $\implies$ $ \dfrac {b^2c^2}{1+a^2} +\dfrac {c^2a^2}{1+b^2}+\dfrac {a^2b^2}{1+c^2} \le \dfrac 14 $
50296
Prove $(-a+b+c)(a-b+c)(a+b-c) \leq abc$, where $a, b$ and $c$ are positive real numbers
6777
Showing $(a^{2}+2)(b^{2}+2)(c^{2}+2)\geq 9(ab+bc+ca)$
65280
If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.
122083
Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$
6621
Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$
132098
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
73277
Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
52157
Prove that $d(n)\leq 2\sqrt{n}$
44247
For $a^3+b^3+c^3=3$ prove that $a^4b+b^4c+c^4a\leq3$