Let $1<a_{1}<a_{2}<\cdots$ be a sequence of positive integers. Show that \[\frac{2^{a_{1}}}{{a_{1}}!}+\frac{2^{a_{2}}}{{a_{2}}!}+\frac{2^{a_{3}}}{{a_{3}}!}+\cdots\] is irrational.

Let $ A$ and $ B$ be two Abelian groups, and define the sum of two homomorphisms $ \eta$ and $ \chi$ from $ A$ to $ B$ by \[ a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ .\] With this addition, the set of homomorphisms from $ A$ to $ B$ forms an Abelian group $ H$. Suppose now that $ A$ is a $ p$-group ( $ p$ a prime number). Prove that in this case $ H$ becomes a topological group under the topology defined by taking the subgroups $ p^kH \;(k\equal{}1,2,...)$ as a neighborhood base of $ 0$. Prove that $ H$ is complete in this topology and that every connected component of $ H$ consists of a single element. When is $ H$ compact in this topology? [L. Fuchs]

Let $f: \mathbb{R}\rightarrow ]0,+\infty[$ be an increasing differentiable function with $\lim_{x\rightarrow+\infty}f(x)=+\infty$ and $f'$ is bounded, and let $F(x)=\int^x_0 f(t) dt$. Define the sequence $(a_n)$ recursively by $a_0=1,a_{n+1}=a_n+\frac1{f(a_n)}$ Define the sequence $(b_n)$ by $b_n=F^{-1}(n)$. Prove that $\lim_{x\rightarrow+\infty}(a_n-b_n)=0$.

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

Evaluate \[\lim_{n\to\infty} \sum_{k=1}^n \frac{[\sqrt{2n^2-k^2}\ ]}{n^2}\] $[x]$ is the greatest integer $\leq x$.

Triangle $ABC$ and a function $f:\mathbb{R}^+\to\mathbb{R}$ have the following property: for every line segment $DE$ from the interior of the triangle with midpoint $M$, the inequality $f(d(D))+f(d(E))\le 2f(d(M))$, where $d(X)$ is the distance from point $X$ to the nearest side of the triangle ($X$ is in the interior of $\triangle ABC$). Prove that for each line segment $PQ$ and each point interior point $N$ the inequality $|QN|f(d(P))+|PN|f(d(Q))\le |PQ|f(d(N))$ holds.

Find $ \lim_{n\to\infty} \int_0^1 x^2|\sin n\pi x|\ dx\ (n\equal{}1,\ 2,\cdots)$.

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

$ \lim_{n\to\infty }\frac{1}{\sqrt[n]{n!}}\left\lfloor \log_5 \sum_{k=2}^{1+5^n} \sqrt[5^n]{k} \right\rfloor $ [i]Taclit Daniela Nadia[/i]

For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that \[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \] \[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]

Consider the sequence $ \left( x_n \right)_{n\ge 1} $ having $ x_1>1 $ and satisfying the equation $$ x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} . $$ Show that this sequence is convergent and find its limit.

An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$. a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole? b) Will the airplne rotate finitely many times around the north pole? If yes how many times?

Compute $ \lim_{n\rightarrow \infty} \int_{0}^{\pi} \frac {\sin{x}}{1 \plus{} \cos^2 nx}dx$ .