Given $x,y\in\mathbb{Q}$ such that $x<y$, $\exists r\in\mathbb{Q} : x<r^2<y$
125164
How to prove the existence of $b$ in $Q$ such that $a<b^2<c$ in $Q$?
92290
For each $x,y,z∈\mathbb N$, if $x<y$ then $x+z<y+z$
29679
Prove: $|x-y|\leq |x|+|y|$
28125
Showing that $1 - \frac{x^2}2\leq\cos x$, $\forall x \in \mathbb{R}$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
68456
Prove that there exists irrational numbers $x$ and $y$ such that $x + y$ is rational, without using subtraction
102478
$X_n\leq Y_n$ implies $\liminf X_n \leq \liminf Y_n$ and $\limsup X_n \leq \limsup Y_n$
130998
Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
62239
For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
47856
$f(x)=ax+b$ for some $a,b\in\mathbb{Q}$ if $f(\mathbb{Q})\subset\mathbb{Q}$ and $f(\mathbb{R-Q})\subseteq \mathbb{R}-\mathbb{Q}$
141529
Proving existence of a minimum
68409
Consider $f(x,y)=\sup_{θ,φ}\{||e^{i θ }x+ e^{i φ }y||^2: θ,φ∈R\}$ in which $x,y\in\mathbb C^n$
20626
Prove that $x^y < y^x$
147892
Proving if $x<y$ then $\sqrt{x} < \sqrt{y}$
109305
Non-existence of natural numbers such that $\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}$
4396
If $A^2\succ B^2$, then necessarily $A\succ B$
47809
If $x,y\in\mathbb Z$, solve $x^2+xy+y^2=x^2y^2$.