SCHWEITZER 2021

Let $n, m \in \mathbb{N}$ ; $a_1,\ldots, a_m \in \mathbb{Z}^n$ . Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied:

• The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace),
• the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$ -th column is $a_i$ as a column vector) is $1$ .

Prove that the equation $$2^x + 5^y - 31^z = n!$$ has only a finite number of non-negative integer solutions $x,y,z,n$ .

Let $I \subset \mathbb{R}$ be a nonempty open interval and let $f: I \cap \mathbb{Q} \to \mathbb{R}$ be a function such that for all $x, y \in I \cap \mathbb{Q}$ , $$4f\left(\frac{3x + y}{4}\right)+ 4f\left(\frac{x + 3y}{4}\right) \le f(x) + 6f\left(\frac{x + y}{2}\right)+ f(y).$$ Show that $f$ can be continuously extended to $I$ .

Let $I$ be a nonempty open subinterval of the set of positive real numbers. For which even $n \in \mathbb{N}$ are there injective function $f: I \to \mathbb{R}$ and positive function $p: I \to \mathbb{R}$ , such that for all $x_1 , \ldots , x_n \in I$ , $$f \left( \frac{1}{2} \left( \frac{x_1+\cdots+x_n}{n}+\sqrt[n]{x_1 \cdots x_n} \right) \right)=\frac{p(x_1)f(x_1)+\cdots+p(x_n)f(x_n)}{p(x_1)+\cdots+p(x_n)}$$ holds?

Let $f(x)=\frac{1+\cos(2 \pi x)}{2}$ , for $x \in \mathbb{R}$ , and $f^n=\underbrace{ f \circ \cdots \circ f}_{n}$ . Is it true that for Lebesgue almost every $x$ , $\lim_{n \to \infty} f^n(x)=1$ ?

Let $f$ and $g$ be $2 \pi$ -periodic integrable functions such that in some neighborhood of $0$ , $g(x) = f(ax)$ with some $a \neq 0$ . Prove that the Fourier series of $f$ and $g$ are simultaneously convergent or divergent at $0$ .

If the binary representations of the positive integers $k$ and $n$ are $k = \sum_{i=0}^{\infty} k_i 2^i$ and $n = \sum_{i=0}^{\infty} n_i 2^i$ , then the logical sum of these numbers is $$k \oplus n =\sum_{i=0}^{\infty} |k_i-n_i|2^i.$$ Let $N$ be an arbitrary positive integer and $(c_k)_{k \in \mathbb{N}}$ be a sequence of complex numbers such that for all $k \in \mathbb{N}$ , $|c_k| \le 1$ . Prove that there exist positive constants $C$ and $\delta$ such that $$\int_{[-\pi,\pi] \times [-\pi, \pi]} \sup_{n<N, n \in \mathbb{N}} \frac{1}{N} \Big| \sum_{k=1}^{n} c_k e^{i(kx+(k \oplus n) y)} \Big| \mathrm d(x,y) \le C \cdot N^{-\delta}$$ holds.

Prove that for a $2$ -dimensional Riemannian manifold there is a metric linear connection with zero curvature if and only if the Gaussian curvature of the Riemannian manifold can be written as the divergence of a vector field.

For a given natural number $n$ , two players randomly (uniformly distributed) select a common number $0 \le j \le n$ , and then each of them independently randomly selects a subset of $\{1,2, \cdots, n \}$ with $j$ elements. Let $p_n$ be the probability that the same set was chosen. Prove that $$\sum_{k=1}^{n} p_k = 2 \log{n} + 2 \gamma - 1 + o(1), (n \to \infty),$$ where $\gamma$ is the Euler constant.

Consider a coin with a head toss probability $p$ where $0 <p <1$ is fixed. Toss the coin several times, the tosses should be independent of each other. Denote by $A_i$ the event that of the $i$ -th, $(i + 1)$ -th, $\ldots$ , the $(i+m-1)$ -th throws, exactly $T$ is the tail. For $T = 1$ , calculate the conditional probability $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ , and for $T = 2$ , prove that $\mathbb{P}(\bar{A_2} \bar{A_3} \cdots \bar{A_m} | A_1)$ has approximation in the form $a+ \tfrac{b}{m} + \mathcal{O}(p^m)$ as $m \to \infty$ .