If $a,b$ are positive rational numbers and $\sqrt a+\sqrt b$ is rational, then both of $\sqrt a,\sqrt b$ are rational numbers
46831
Show that $ a，b, \sqrt{a}+ \sqrt{b} \in\mathbb Q \implies \sqrt{a},\sqrt{b} \in\mathbb Q $
112171
Proposition: $\sqrt{x + \sqrt{x + \sqrt{x + ...}}} = \frac{1 + \sqrt{1 + 4x}}{2}$.
69575
If $ad$ and $bc$ are odd and even, respectively, then prove that $ax^3+bx^2+cx+d$ has an irrational root.
64654
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
92300
For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$
28347
Prove that $\sqrt 2 + \sqrt 3$ is irrational
63931
When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.
71012
If $a, b$ are positive integers such that $a|b^2, b^2|a^3,....$ then $a=b$.
96217
If a and b are non-negative real numbers then demonstrate inequality
32848
Prove that the product of a non-zero rational and irrational number is irrational.
134857
Find $(a,b)$ if $x^2-bx+a = 0, x^2-ax+b = 0$ both have distinct positive integers roots
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
138216
$a,b,c$ are positive real numbers such that, $a+b+c\ge abc$. Prove that $a^2+b^2+c^2\ge \sqrt{3}abc$
3955
Square roots -- positive and negative
127296
prove that a|b and b|a iff a=+-b
53660
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q  \implies  \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $
85977
Show that if $\sqrt{a}+\sqrt{b}\neq 0$, then $\mathbb{Q}(\sqrt{a}+\sqrt{b}) = \mathbb{Q}(\sqrt{a},\sqrt{b}), \forall a,b\in\mathbb{Q}$