Found problems: 8
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
Prove that if X is a compact $T_2$ space, and X has density d(X), then $X^3$ contains a discrete subspace of cardinality $d(X)$.
note: $d(X)$ is the smallest cardinality of a dense subspace of X.
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
Let $ \left( c_n \right)_{n\ge 1} $ be a sequence of real numbers. Prove that the sequences $ \left( c_n\sin n \right)_{n\ge 1} ,\left( c_n\cos n \right)_{n\ge 1} $ are both convergent if and only if $ \left( c_n \right)_{n\ge 1} $ converges to $ 0. $
[i]Mihai Piticari[/i] and [i]Vladimir Cerbu[/i]
Let $k$ be a positive integer congruent to $1$ modulo $4$ which is not a perfect square and let $a=\frac{1+\sqrt{k}}{2}$.
Show that $\{\left \lfloor{a^2n}\right \rfloor-\left \lfloor{a\left \lfloor{an}\right \rfloor}\right \rfloor : n \in \mathbb{N}_{>0}\}=\{1 , 2 , \ldots ,\left \lfloor{a}\right \rfloor\}$.
Does there exist a piecewise linear continuous function $f:\mathbb{R}\to \mathbb{R}$ such that for any two-way infinite sequence $a_n\in[0,1]$, $n\in\mathbb{Z}$, there exists an $x\in\mathbb{R}$ with
\[
\limsup_{K\to \infty} \frac{\#\{k\le K\,:\, k\in\mathbb{N},f^k(x)\in[n,n+1)\}}{K}=a_n
\]
for all $n\in\mathbb{Z}$, where $f^k=f\circ f\circ \dots\circ f$ stands for the $k$-fold iterate of $f$?
A truncated cone has the bigger base of radius $r$ centimetres and the generatrix makes an angle, with that base, whose tangent equals $m$. The truncated cone is constructed of a material of density $d$ (g/cm$^3$) and the smaller base is covered by a special material of density $p$ (g/cm$^2$). Which is the height of the truncated cone that maximizes the total mass?