Let $ K$ be a convex cone in the $ n$-dimensional real vector space $ \mathbb{R}^n$, and consider the sets $ A\equal{}K \cup (\minus{}K)$ and $ B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ($ 0$ is the origin). Show that one can find two subspaces in $ \mathbb{R}^n$ such that together they span $ \mathbb{R}^n$, and one of them lies in $ A$ and the other lies in $ B$. [i]J. Szucs[/i]

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. [i]Proposed by Péter Nagy Tibor[/i]

Let $ f(n)$ denote the maximum possible number of right triangles determined by $ n$ coplanar points. Show that \[ \lim_{n\rightarrow \infty} \frac{f(n)}{n^2}\equal{}\infty \;\textrm{and}\ \lim_{n\rightarrow \infty}\frac{f(n)}{n^3}\equal{}0 .\] [i]P. Erdos[/i]

Let $ f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $ N_k$ of natural numbers such that \[ f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\ .\] Prove that $ f$ is identically $ 1$. [i]I. Katai[/i]

Let $ f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $ \mathbb{R}^3$ for which the inequality $ f(x_0,0,0) \leq \varepsilon^2$ holds for some $ 0\leq x_0 \leq 1$ and $ 0<\varepsilon<(1\minus{}x_0)^2$. Prove that $ f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $ (1\minus{}3\varepsilon^{1/4}).$ [i]P. Turan[/i]

Assume that a face of a convex polyhedron $ P$ has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of $ P$ and passes through all vertices. [i]L .Lovasz[/i]

Let $ H$ be an arbitrary subgroup of the diffeomorphism group $ \mathsf{Diff}^\infty(M)$ of a differentiable manifold $ M$. We say that an $ \mathcal C^\infty$-vector field $ X$ is [i]weakly tangent[/i] to the group $ H$, if there exists a positive integer $ k$ and a $ \mathcal C^\infty$-differentiable map $ \varphi \mathrel{: } \mathord{]} \minus{} \varepsilon,\varepsilon\mathord{[}^k\times M\to M$ such that (i) for fixed $ t_1,\dots,t_k$ the map \[ \varphi_{t_1,\dots,t_k} : x\in M\mapsto \varphi(t_1,\dots,t_k,x)\] is a diffeomorphism of $ M$, and $ \varphi_{t_1,\dots,t_k}\in H$; (ii) $ \varphi_{t_1,\dots,t_k}\in H \equal{} \mathsf{Id}$ whenever $ t_j \equal{} 0$ for some $ 1\leq j\leq k$; (iii) for any $ \mathcal C^\infty$-function $ f: M\to \mathbb R$ \[ X f \equal{} \left.\frac {\partial^k(f\circ\varphi_{t_1,\dots,t_k})}{\partial t_1\dots\partial t_k}\right|_{(t_1,\dots,t_k) \equal{} (0,\dots,0)}.\] Prove, that the commutators of $ \mathcal C^\infty$-vector fields that are weakly tangent to $ H\subset \textsf{Diff}^\infty(M)$ are also weakly tangent to $ H$.

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

$F$* is the multiplicative group of the field $F$. $F$* is of finitely generated. Is it true that $F$* is cyclic? Additional question: (wasn’t at the olympiad) $K$* is the multiplicative group of the field $K$. $L \subseteq $$K$* is a finitely generated subgroup. Is it true that $L$ is cyclic?

Is it true that on any surface homeomorphic to an open disc there exist two congruent curves homeomorphic to a circle?

Let $ X_1, \ldots ,X_n$ be $ n$ points in the unit square ($ n>1$). Let $ r_i$ be the distance of $ X_i$ from the nearest point (other than $ X_i$). Prove that the inequality \[ r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4.\] [i]L. Fejes-Toth, E. Szemeredi[/i]

Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic. [i]F. Karteszi[/i]

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

Let $ p$ be a prime and let \[ l_k(x,y)\equal{}a_kx\plus{}b_ky \;(k\equal{}1,2,...,p^2)\ .\] be homogeneous linear polynomials with integral coefficients. Suppose that for every pair $ (\xi,\eta)$ of integers, not both divisible by $ p$, the values $ l_k(\xi,\eta), \;1\leq k\leq p^2 $, represent every residue class $ \textrm{mod} \;p$ exactly $ p$ times. Prove that the set of pairs $ \{(a_k,b_k): 1\leq k \leq p^2 \}$ is identical $ \textrm{mod} \;p$ with the set $ \{(m,n): 0\leq m,n \leq p\minus{}1 \}.$

Let $ \mathcal{F}$ be a family of subsets of a ground set $ X$ such that $ \cup_{F \in \mathcal{F}}F=X$, and (a) if $ A,B \in \mathcal{F}$, then $ A \cup B \subseteq C$ for some $ C \in \mathcal{F};$ (b) if $ A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},$ and $ A_0 \subset A_1 \subset...,$ then, for some $ k \geq 0, \;A_n \cap B=A_k \cap B$ for all $ n \geq k$. Show that there exist pairwise disjoint sets ${ X_{\gamma} \;( \gamma \in \Gamma}\ )$, with $ X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},$ such that every $ X_{\gamma}$ is contained in some member of $ \mathcal{F}$, and every element of $ \mathcal{F}$ is contained in the union of finitely many $ X_{\gamma}$'s. [i]A. Hajnal[/i]

Let $ D$ be a convex subset of the $ n$-dimensional space, and suppose that $ D'$ is obtained from $ D$ by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of $ D$ and $ D'$ is $ 1$, and $ D \cap D'\not\equal{} \emptyset .$ Determine the supremum of the volume of the convex hull of $ D \cup D'$ taken for all such pairs of sets $ D,D'$. [i]L. Fejes-Toth, E. Makai[/i]

Let $ \mathcal{H}$ be the class of all graphs with at most $ 2^{\aleph_0}$ vertices not containing a complete subgraph of size $ \aleph_1$. Show that there is no graph $ H \in \mathcal{H}$ such that every graph in $ \mathcal{H}$ is a subgraph of $ H$. [i]F. Galvin[/i]

Let $ \sigma(S_n,k)$ denote the sum of the $ k$th powers of the lengths of the sides of the convex $ n$-gon $ S_n$ inscribed in a unit circle. Show that for any natural number greater than $ 2$ there exists a real number $ k_0$ between $ 1$ and $ 2$ such that $ \sigma(S_n,k_0)$ attains its maximum for the regular $ n$-gon. [i]L. Fejes Toth[/i]

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 \[ \displaystyle \partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n\] then ${x}$ is a point on ${g(a,b)}$ and conversely. [i]Proposed by Lajos Tamássy and Dávid Kertész[/i]

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

For which cardinalities $ \kappa$ do antimetric spaces of cardinality $ \kappa$ exist? $ (X,\varrho)$ is called an $ \textit{antimetric space}$ if $ X$ is a nonempty set, $ \varrho : X^2 \rightarrow [0,\infty)$ is a symmetric map, $ \varrho(x,y)\equal{}0$ holds iff $ x\equal{}y$, and for any three-element subset $ \{a_1,a_2,a_3 \}$ of $ X$ \[ \varrho(a_{1f},a_{2f})\plus{}\varrho(a_{2f},a_{3f}) < \varrho(a_{1f},a_{3f})\] holds for some permutation $ f$ of $ \{1,2,3 \}$. [i]V. Totik[/i]

Give an example of ten different noncoplanar points $ P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $ 3$-space such that connecting each $ P_i$ to each $ Q_j$ by a rigid rod results in a rigid system. [i]L. Lovasz[/i]

Let $ V$ be a bounded, closed, convex set in $ \mathbb{R}^n$, and denote by $ r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $ V$). Show that $ r$ is the only real number with the following property: for any finite number of points in $ V$, there exists a point in $ V$ such that the arithmetic mean of its distances from the other points is equal to $ r$. [i]Gy. Szekeres[/i]

Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]

[b]a)[/b] prove that the function $\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ that is defined on the area $Re(s)>1$, is an analytic function. [b]b)[/b] prove that the function $\zeta(s)-\frac{1}{s-1}$ can be spanned to an analytic function over $\mathbb C$. [b]c)[/b] using the span of part [b]b[/b] show that $\zeta(1-n)=-\frac{B_n}{n}$ that $B_n$ is the $n$th bernoli number that is defined by generating function $\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}$.