SCHWEITZER 2013

Let $q$ be a positive integer. Prove there exists a constant $C_q$ such that the following inequality holds for any finite set $A$ of integers: $$|A+qA|\ge (q+1)|A|-C_q.$$

Proposed by Antal Balog.

Prove there exists a constant $k_0$ such that for any $k\ge k_0$ , the equation $$a^{2n}+b^{4n}+2013=ka^nb^{2n}$$ has no positive integer solutions $a,b,n$ .

Proposed by István Pink.

Find for which positive integers $n$ the $A_n$ alternating group has a permutation which is contained in exactly one $2$ -Sylow subgroup of $A_n$ .

Proposed by Péter Pál Pálfy

Let $A$ be an Abelian group with $n$ elements. Prove that there are two subgroups in $\text{GL}(n,\Bbb{C})$ , isomorphic to $S_n$ , whose intersection is isomorphic to the automorphism group of $A$ .

Proposed by Zoltán Halasi

A subalgebra $\mathfrak{h}$ of a Lie algebra $\mathfrak g$ is said to have the $\gamma$ property with respect to a scalar product ${\langle \cdot,\cdot \rangle}$ given on ${\mathfrak g}$ if ${X \in \mathfrak{h}}$ implies ${\langle [X,Y],X\rangle =0}$ for all ${Y \in \mathfrak g}$ . Prove that the maximum dimension of ${\gamma}$ -property subalgebras of a given ${2}$ step nilpotent Lie algebra with respect to a scalar product is independent of the selection of the scalar product.

Proposed by Péter Nagy Tibor

Let ${\mathcal A}$ be a ${C^{\ast}}$ algebra with a unit element and let ${\mathcal A_+}$ be the cone of the positive elements of ${\mathcal A}$ (this is the set of such self adjoint elements in ${\mathcal A}$ whose spectrum is in ${[0,\infty)}$ . Consider the operation $$x \circ y =\sqrt{x}y\sqrt{x},\ x,y \in \mathcal A_+$$ Prove that if for all ${x,y \in \mathcal A_+}$ we have $$(x\circ y)\circ y = x \circ (y \circ y),$$ then ${\mathcal A}$ is commutative.

Proposed by Lajos Molnár

Suppose that ${f: \Bbb{R} \rightarrow \Bbb{R}}$ is an additive function $($ that is ${f(x+y) = f(x)+f(y)}$ for all ${x, y \in \Bbb{R}})$ for which ${x \mapsto f(x)f(\sqrt{1-x^2})}$ is bounded of some nonempty subinterval of ${(0,1)}$ . Prove that ${f}$ is continuous.

Proposed by Zoltán Boros

Let ${f : \Bbb{R} \rightarrow \Bbb{R}}$ be a continuous and strictly increasing function for which $$f^{-1}\left(\frac{f(x)+f(y)}{2}\right)(f(x)+f(y)) =(x+y)f\left(\frac{x+y}{2}\right)$$ for all ${x,y \in \Bbb{R}} ({f^{-1}}$ denotes the inverse of ${f})$ . Prove that there exist real constants ${a \neq 0}$ and ${b}$ such that ${f(x)=ax+b}$ for all ${x \in \Bbb{R}}.$ Proposed by Zoltán Daróczy

Prove that there is a function ${f: (0,\infty) \rightarrow (0,\infty)}$ which is nowhere continuous and for all ${x,y \in (0,\infty)}$ and any rational ${\alpha}$ we have $$f\left( \left(\frac{x^\alpha+y^\alpha}{2}\right)^{\frac{1}{\alpha}}\right)\leq \left(\frac{f(x)^\alpha +f(y)^\alpha }{2}\right)^{\frac{1}{\alpha}}.$$ Is there such a function if instead the above relation holds for every ${x,y \in (0,\infty)}$ and for every irrational ${\alpha}?$ Proposed by Maksa Gyula and Zsolt Páles

Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$ . Suppose that for all ${a \in \Bbb{R}^n}$ , the Riemannian distance with respect to ${a}, {\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$ . Prove that if for a point ${x \neq a,b}$ we have $$\partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n$$ then ${x}$ is a point on ${g(a,b)}$ and conversely.

Proposed by Lajos Tamássy and Dávid Kertész

[list] (a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value. (b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere. [/list]

Proposed by Tran Quoc Binh

There are ${n}$ tokens in a pack. Some of them (at least one, but not all) are white and the rest are black. All tokens are extracted randomly from the pack, one by one, without putting them back. Let ${X_i}$ be the ratio of white tokens in the pack before the ${i^{\text{th}}}$ extraction and let $$T =\max \{ |X_i-X_j| : 1 \leq i \leq j \leq n\}.$$ Prove that ${\Bbb{E}(T) \leq H(\Bbb{E}(X_1))},$ where ${H(x)=-x\ln x -(1-x)\ln(1-x)}.$ Proposed by Tamás Móri