Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. ...

1. Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements.
2. Let $k\geq 1$ and let $I_{1},\dots, I_{k}$ be non-degenerate subintervals of the interval $[0, 1]$. Prove that $\sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2}$ where the summation is over all pairs $(i, j)$ of indices such that $I_i\cap I_j\neq \emptyset$.
3. We have $4n + 5$ points on the plane, no three of them are collinear. The points are colored with two colors. Prove that from the points we can form $n$ empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occurring as vertices of the $n$ triangles have the same color.
4. For a positive integer $n$, define $f(n)$ to be the number of sequences $(a_1,a_2,\dots,a_k)$ such that $a_1a_2\cdots a_k=n$ where $a_i\geq 2$ and $k\ge 0$ is arbitrary. Also we define $f(1)=1$. Now let $\alpha>1$ be the unique real number satisfying $\zeta(\alpha)=2$, i.e $\sum_{n=1}^{\infty}\frac{1}{n^\alpha}=2$. Prove that
a) $\displaystyle{\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\alpha).}$
b) There is no real number $\beta<\alpha$ such that $\displaystyle{\sum_{j=1}^{n}f(j)=\mathcal{O}(n^\beta).}$
5. Let $\alpha$ be a non-real algebraic integer of degree two, and let $\mathbb{P}$ be the set of irreducible elements of the ring $\mathbb{Z}[ \alpha]$. Prove that $\sum_{p\in \mathbb{P}}^{{}}\frac{1}{|p|^{2}}=\infty$
6. Let $\rho:G\to GL(V)$ be a representation of a finite $p$-group $G$ over a field of characteristic $p$. Prove that if the restriction of the linear map $\sum_{g\in G} \rho(g)$ to a finite dimensional subspace $W$ of $V$ is injective, then the subspace spanned by the subspaces $\rho(g)W$ $(g\in G)$ is the direct sum of these subspaces.
7. Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?
8. Let $n\ge 1$ be a fixed integer. Calculate the distance $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ with real coefficients and $f$ runs over functions $f(x)= \sum_{k=n}^{\infty} c_k x^k$ defined on the closed interval $[0,1]$ , where $c_k \ge 0$ and $\sum_{k=n}^{\infty} c_k=1$.
9. Let $\rho:\mathbb{R}^n\to \mathbb{R}$, $\rho(\mathbf{x})=e^{-||\mathbf{x}||^2}$, and let $K\subset \mathbb{R}^n$ be a convex body, i.e., a compact convex set with nonempty interior. Define the barycenter $\mathbf{s}_K$ of the body $K$ with respect to the weight function $\rho$ by the usual formula $\mathbf{s}_K=\frac{\int_K\rho(\mathbf{x})\mathbf{x}d\mathbf{x}}{\int_K\rho(\mathbf{x})d\mathbf{x}}.$ Prove that the translates of the body $K$ have pairwise distinct barycenters with respect to $\rho$.
10. To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.
11. Let $U$ be a random variable that is uniformly distributed on the interval $[0,1]$, and let $S_n= 2\sum_{k=1}^n \sin(2kU\pi).$ Show that, as $n\to \infty$, the limit distribution of $S_n$ is the Cauchy distribution with density function $f(x)=\frac1{\pi(1+x^2)}$.

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