SCHWEITZER 2010

Let $p$ be prime. Denote by $N (p)$ the number of integers $x$ for which $1 \leq x \leq p$ and $$x ^ {x} \equiv 1 (\bmod p)$$ Prove that there exist numbers $c <1/2$ and $p_ {0}> 0$ such that $$N (p) \leq p ^ {c}$$ if $p \ge p_ {0}$ .

Let $G$ be a countably infinite, $d$ -regular, connected, vertex-transitive graph. Show that there is a complete pairing in $G$ .

Let $A_i,i=1,2,\dots,t$ be distinct subsets of the base set $\{1,2,\dots,n\}$ complying to the following condition $$A_ {i} \cap A_ {k} \subseteq A_ {j}$$ for any $1 \leq i <j <k \leq t.$ Find the maximum value of $t.$ Thanks @dgrozev

Prove that if $n \geq 2$ and $I_ {1}, I_ {2}, \ldots, I_ {n}$ are idealized in a unit-element commutative ring such that any nonempty $H \subseteq \{ 1,2, \dots, n \}$ then if $\sum_ {h \in H} I_ {h}$ Is ideal $$I_ {2} I_ {3} I_ {4} \dots I_ {n} + I_ {1} I_ {3} I_ {4} \dots I_ {n} + \dots + I_ {1} I_ {2} \dots I_ {n-1}$$ also Is ideal.

Given the vectors $v_ {1}, \dots, v_ {n}$ and $w_ {1}, \dots, w_ {n}$ in the plane with the following properties: for every $1 \leq i \leq n$ , $\left | v_{i} -w_{i} \right | \leq 1,$ and for every $1 \leq i <j \leq n$ , $\left | v_{i} -v_{j} \right | \ge 3$ and $v_{i} -w_ {i} \ne v_ {j} -w_ {j}$ . Prove that for sets $V = \left \{v_ {1}, \dots, v_{n } \right \}$ and $W = \left \{w_ {1}, \dots, w_ {n} \right \}$ , the set of $V + (V \cup W)$ must have at least $cn^{3/2}$ elements ,for some universal constant $c>0$ .

Is there a continuous function $f: \mathbb {R} ^ {2} \rightarrow \mathbb {R}$ for every $d \in \mathbb {R}$ we have $g_{m,d}(x) = f (x, m x + d)$ is strictly monotonic on $\mathbb {R}$ if $m \ge 0,$ and not monotone on any non-empty open interval if $m <0?$

Is there any sequence $(a_n)_{n=1}^{\infty}$ of non-negative numbers, for which $\sum_{n=1}^{\infty} a_n^2<\infty$ , but $\sum_{n=1}^{\infty}\left(\sum_{k=1}^{\infty}\frac{a_{kn}}{k} \right)^2=\infty$ ?

Let $D \subset \mathbb {R} ^ {2}$ be a finite Lebesgue measure of a connected open set and $u: D \rightarrow \mathbb {R}$ a harmonic function. Show that it is either a constant $u$ or for almost every $p \in D$ $$f ^ {\prime} (t) = (\operatorname {grad} u) (f (t)), f (0) = p$$ has no initial value problem(differentiable everywhere) solution to $f:[0,\infty) \rightarrow D$ .

For each $M$ m-dimensional closed $C^{\infty}$ set , assign a $G(m)$ in some euclidean space $\mathbb{R}^{q}$ . Denote by $\mathbb{R} \mathbb{P}^{q}$ a $q$ -dimensional real projecive space. A $G(M) \subseteq \times \mathbb{R} \mathbb{P}^{q}$ . The set consists of $(x,e)$ pairs for which $x \in M \subseteq \mathbb {P}^{q}$ and $e \subseteq \mathbb {R}^{q+1}= \mathbb{R}^{q} \times \mathbb{R}$ and $\mathrm{a} (0, \ldots,0,1) \in \mathbb{R}^{q+1}$ in a stretched $(m+1)$ -dimensional linear subspace. Prove that if $N$ is a $n$ -dimensional closed set $C^{\infty}$ , then $P=G(M \times M)$ and $Q=G(M) \times G(N)$ are cobordant , that is, there exists a $(2m+2n+1)$ -dimensional compact , flanged set $C^{\infty}$ with a disjoint union of $P$ and $Q$ .

Consider the space $\{0,1 \} ^{N}$ with the product topology (where $\{0,1 \}$ is a discrete space). Let $T: \{0,1 \} ^ {\mathbb {N}} \rightarrow \{0,1 \} ^ {\mathbb {N}}$ be the left-shift, ie $(Tx) (n) = x (n+1)$ for every $n \in \mathbb {N}$ . Can a finite number of Borel sets be given: $B_ {1}, \ldots, B_ {m} \subset \{0,1 \} ^ {N}$ such that $$\left \{T ^ {i} \left (B_ {j} \right) \mid i \in \mathbb {N}, 1 \leq j \leq m \right \}$$ the $\sigma$ -algebra generated by the set system coincides with the Borel set system?

For problem 11 , i couldn't find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful . [tip=see hungarian]Az $X$ ́es $Y$ valo ́s ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az $f(X)$ ́es $g(Y )$ v ́altoz ́ok korrela ́cio ́j ́anak szupr ́emuma az olyan $f$ ́es $g$ Borel m ́erheto ̋, $\mathbb{R} \to \mathbb{R}$ fu ̈ggv ́enyeken, amelyekre $f(X)$ ́es $g(Y)$ v ́eges sz ́ora ́su ́. Legyen U a $[0,2\pi]$ interval- lumon egyenletes eloszl ́asu ́ val ́osz ́ınu ̋s ́egi v ́altozo ́, valamint n ́es m pozit ́ıv eg ́eszek. Sz ́am ́ıtsuk ki $\sin(nU)$ ́es $\sin(mU)$ maxim ́alkorrela ́ci ́oja ́t. [/tip] Edit: