SCHWEITZER 2007

Prove that there exist subfields of $\mathbb R$ that are a) non-measurable and b) of measure zero and continuum cardinality.

(translated by Miklós Maróti)

We partition the $n$ -element subsets of an $n^2+n-1$ -element set into two classes. Prove that one of the classes contains $n$ -many pairwise disjunct sets.

(translated by Miklós Maróti)

Denote by $\omega (n)$ the number of prime divisors of the natural number $n$ (without multiplicities). Let $$F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)$$ Prove that $G(x)-F(x)\to \infty$ as $x\to\infty$ .

(translated by Miklós Maróti)

Let $p$ be a prime number and $a_1, \ldots, a_{p-1}$ be not necessarily distinct nonzero elements of the $p$ -element $\mathbb Z_p \pmod{p}$ group. Prove that each element of $\mathbb Z_p$ equals a sum of some of the $a_i$ 's (the empty sum is $0$ ).

(translated by Miklós Maróti)

Let $D=\{ (x,y) \mid x>0, y\neq 0\}$ and let $u\in C^1(\overline {D})$ be a bounded function that is harmonic on $D$ and for which $u=0$ on the $y$ -axis. Prove that $u$ is identically zero.

(translated by Miklós Maróti)

For which subsets $A\subset \mathbb R$ is it true that whenever $0\leq x_0 < x_1 < \cdots < x_n\leq 1$ , $n=1,2, \ldots$ , then there exist $y_j\in A$ numbers, such that $y_{j+1}-y_j>x_{j+1}-x_j$ for all $0\leq j < n$ .

(translated by Miklós Maróti)

Prove that there exist natural numbers $n_k, m_k, k=0,1,2,\ldots$ , such that the numbers $n_k+m_k, k=1,2,\ldots$ are pairwise distinct primes and the set of linear combination of the polynomials $x^{n_k}y^{m_k}$ is dense in $C([0,1] \times [0,1])$ under the supremum norm.

(translated by Miklós Maróti)

For an $A=\{ a_i\}^{\infty}_{i=0}$ sequence let $SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}$ be the sequence of partial sums of the $a_0+a_1+\ldots$ series. Does there exist a non-identically zero sequence $A$ such that all of the sequences $A, SA, SSA, SSSA, \ldots$ are convergent?

(translated by Miklós Maróti)

Let $A$ and $B$ be two triangles on the plane such that the interior of both contains the origin and for each circle $C_r$ centered at the origin $|C_r \cap A|=|C_r\cap B|$ (where $|\cdot |$ is the arc-length measure). Prove that $A$ and $B$ are congruent. Does this statement remain true if the origin is on the border of $A$ or $B$ ?

(translated by Miklós Maróti)

Let $\zeta_1, \zeta_2,\ldots$ be identically distributed, independent real-valued random variables with expected value $0$ . Suppose that the $\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)$ logarithmic moment-generating function always exists for $\lambda\in\mathbb R$ ( $\mathbb E$ is the expected value). Furthermore, let $G\colon\mathbb R \rightarrow \mathbb R$ be a function such that $G(x)\leq \min (|x|, x^2)$ . Prove that for small $\gamma >0$ the following sequence is bounded: $$\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}$$ (translated by j___d)