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Problems of the Miklos Schweitzer Competition 2014

November 10, 2014 5 comments

Problem 1. Let {n} be a positive integer. Let {\mathcal{F}} be a familiy of sets that contains more than half of all subsets of an {n}-element set {X}. Prove that from {\mathcal{F}} we can select {\lceil \log_2 n\rceil+1 } sets that form a separating family of {X}, i.e., for any two distinct elements of {X} there is a selected set containing exactly one of the two elements.

Problem 2. let {k \geq 1} and let {I_1,...,I_k} be non-degenerate subintervals of the interval {[0,1]}. Prove that

\displaystyle \sum \frac{1}{|I_i \cup I_j|} \geq k^2,

where the summation is over all pairs of indices {(i,j)} such that {I_i} and {I_j} are not disjoint.

Problem 3. We have {4n+5} points in the plane, no three of them collinear. The points are colored with two colors. Prove that from the points we can form {n} empty triangles (they have no colored points in their interiors) with pairwise disjoint interiors, such that all points occuring as vertices of the {n} triangles have the same color.

Problem 4. For a positive integer {n}, let {f(n)} be the number of sequences {a_1,...,a_k} of positive integers such that {a_i \geq 2} and {a_1...a_k = n} for {k \geq 1}. We make the convention {f(1)=1}. Let {\alpha} be the unique real number greater than {1} such that {\sum_{n=1}^\infty n^{-\alpha}=2}. Prove that

  • (i) { \sum_{ k = 1}^n f(k)= O(n^\alpha)}.
  • (ii) There exists no number {\beta<\alpha} such that {f(n)=O(n^\beta)}.

Problem 5. Let {\alpha} be a non-real algebraic integer of degree two, and let {P} be the set of irreducible elements of the ring {\Bbb{Z}[\alpha]}. Prove that

\displaystyle \sum_{ p \in P} \frac{1}{|p|^2} = \infty.

Problem 6. Let {\rho : G \rightarrow GL(V)} be a representation of a finite {p}-group {G} over a field of characteristic {p}. Prove that if the restriction of the linear map {\sum_{ g \in G} \rho(g)} to a finite dimensional subspace {W} of {V} is injective, then the subspace spanned by the subspaces {\rho(g)W} ({g \in G}) is the direct sum of these subspaces.

Problem 7. Lef {f: \Bbb{R} \rightarrow \Bbb{R}} be a continuous function and let {g: \Bbb{R} \rightarrow \Bbb{R}} be arbitrary. Suppose that the Minkowski sum of the graph of {f} and the graph of {g} (i.e. the set {\{(x+y,f(x)+g(y) : x,y \in \Bbb{R}\}} has Lebesgue measure zero. Does it follow then that the function {f} is of the form {f(x)=ax+b}, with suitable constants {a,b \in \Bbb{R}}?

Problem 8. Let {n \geq 1} be a fixed integer. Calculate the distance

\displaystyle \inf_{p,f} \max_{0 \leq x \leq 1} |f(x)-p(x)|,

where {p} runs over polynomials of degree less than {n} with real coefficients and {f} runs over functions of the form

\displaystyle f(x) = \sum_{ k = n}^\infty c_kx^k

defined on the closed interval {[0,1]}, where {c_k\geq 0} and {\sum_{k=n}^\infty c_k =1}.

Problem 9. Let {\rho : \Bbb{R}^n \rightarrow \Bbb{R},\ \rho(x)=e^{-\|x\|^2}}, and let {K \subset \Bbb{R}^n} be a convex body, i.e. a compact convex set with nonempty interior. Define the barycenter {s_K} of the body {K} with respect to the weight function {\rho} by the usual formula

\displaystyle s_K = \frac{\int_K \rho(x) x dx}{\int_K \rho(x)dx}.

Prove that the translates of the body {K} have pairwise distinct barycenters with respect to {\rho}.

Problem 10. To each vertex of a given triangulation of the two dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

Problem 11. Let {U} be a random variable that is uniformly distributed on the interval {[0,1]}, and let

\displaystyle S_n = 2\sum_{k=1}^n \sin(2kU\pi).

Show that, as {n \rightarrow \infty}, the limit distribution of {S_n} is the Cauchy distribution with density function {f(x) =\frac{1}{\pi(1+x^2)}}.

Source: http://www.bolyai.hu/SCH_angol_2014.pdf

Miklos Schweitzer 2013 Problem 7

November 22, 2013 1 comment

Problem 7. 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.

Miklos Schweitzer 2013

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Miklos Schweitzer 2013

November 9, 2013 2 comments

Problem 1. Let {q \geq 1} be an integer. Prove that there exists an integer {C_q} such that for any finite set {A} containing only integers we have

\displaystyle |A+qA| \geq (q+1)|A|-C_q.

({A+qA} is the set of integers of the form {a+qa'} where {a,a' \in A})

Problem 2. Prove that there is a constant {k_0} such that the diophantine equation

\displaystyle a^{2n}+b^{4n}+2013=ka^n b^{2n}

has no positive integer solutions {(a,b,n)} for {k \geq k_0}. (here I guess k needs to be an integer, or else the conclusion is clearly false)

Problem 3. What are the numbers {n} for which the {A_n} alternating group has a permutation which is contained in exactly one {2}-Sylow subgroup of {A_n}?

Problem 4. Let {A} be an Abelian group with {n} elements. Prove that there are two subgroups in {GL(n,\Bbb{C})}, isomorphic to {S_n}, whose intersection is isomorphic to the automorphism group of {A}.

Problem 5. A subalgebra {\mathfrak h} of a Lie algebra {\mathfrak g} is said to have tha {\gamma} property with respect to a scalar product {\langle .,.\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.

Problem 6. Let {\mathcal A} be a {C^*} 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

\displaystyle 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

\displaystyle (x\circ y)\circ y = x \circ (y \circ y),

then {\mathcal A} is commutative.

Problem 7. 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. [Solution]

Problem 8. Let {f : \Bbb{R} \rightarrow \Bbb{R}} be a continuous and strictly increasing function for which

\displaystyle 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}}. [Solution]

Problem 9. 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

\displaystyle 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}.

Problem 10. 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..n

then {x} is a point on {g(a,b)} and conversely.

Problem 11. (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.

Problem 12. 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}-th extraction and let

\displaystyle 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)}.

(Thanks again to Eles Andras for the translation of the problems from Hungarian to English)

Commutators and the identity matrix – Miklos Schweitzer 2012 Problem 6

January 8, 2013 1 comment

Suppose {A,B,C} are complex {n\times n} matrices such that {[A,B]=C,\ [B,C]=A} and {[C,A]=B}, where {[X,Y]=XY-YX} is the usual commutator. Prove that {e^{4\pi A}} is the identity matrix.

Miklos Schweitzer 2012 Problem 6

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Miklos Schweitzer 2012 Problems

January 4, 2013 3 comments

1. Is there any real number {\alpha} for which there exist two functions {f,g: \Bbb{N} \rightarrow \Bbb{N}} such that

\displaystyle \alpha=\lim_{n \rightarrow \infty} \frac{f(n)}{g(n)},

but the function which associates to {n} the {n}-th decimal digit of {\alpha} is not recursive?

2. Call a subset {A} of the cyclic group {(\Bbb{Z}_n,+)} rich if for all {x,y \in \Bbb{Z}_n} there exists {r \in \Bbb{Z}_n} such that {x-r,x+r,y-r,y+r} are all in {A}. For what {\alpha} is there a constant {C_\alpha>0} such that for each odd positive integer {n}, every rich subset {A \subset \Bbb{Z}_n} has at least {C_\alpha n^\alpha} elements?

3. Prove that if a {k}-chromatic graph’s edges are coloured in two colors in any way, there is a subtree with {k} vertices and edges of the same color.

4. Let {K} be a convex shape in the {n} dimensional space, having unit volume. Let {S \subset K} be a Lebesgue measurable set with measure at least {1-\varepsilon}, where {0<\varepsilon<1/3}. Prove that dilating {K} from its centroid by the ratio of {2\varepsilon \ln(1/\varepsilon)}, the shape obtained contains the centroid of {S}.

5. Let {V_1,V_2,V_3,V_4} be four dimensional linear subspaces in {\Bbb{R}^8} such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace {W} in {\Bbb{R}^8} such that all four vector spaces {W\cap V_i} are two dimensional.

6. Let {A,B,C} be matrices with complex elements such that {[A,B]=C,\ [B,C]=A} and {[C,A]=B}, where {[X,Y]} denotes the {XY-YX} commutator of the matrices. Prove that {e^{4 \pi A}} is the identity matrix. [Solution]

7. Let {\Gamma} be a simple curve, lying inside a circle of radius {r}, rectifiable and of length {l}. Prove that if {l > kr\pi}, then there exists a circle of radius {r} which intersects {\Gamma} in at least {k+1} distinct points.

8. For any function {f: \Bbb{R}^2\rightarrow \Bbb{R}} consider the function {\Phi_f:\Bbb{R}^2\rightarrow [-\infty,\infty]} for which {\Phi_f(x,y)=\limsup_{ z \rightarrow y} f(x,z)} for any {(x,y) \in \Bbb{R}^2}.

a) Is it true that if {f} is Lebesgue measurable then {\Phi_f} is also Lebesgue measurable?

b) Is it true that if {f} is Borel measurable then {\Phi_f} is also Borel measurable?

9. Let {D} be the complex unit disk {D=\{x \in \Bbb{C}: |z|<1\}}, and {0<a<1} a real number. Suppose that {f:D \rightarrow \Bbb{C}\setminus \{0\}} is a holomorphic function such that {f(a)=1} and {f(-a)=-1}. Prove that

\displaystyle \sup_{z \in D} |f(z)| \geq \exp\left(\frac{1-a^2}{4a}\pi\right)

10. Let {K} be a knot in the {3}-dimensional space (that is a differentiable injection of a circle into {\Bbb{R}^3}), and {D} be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of {D} in black and color the diagram {D} in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define {\Gamma_B(D)} the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.

a) Determine all knots having a diagram {D} such that {\gamma_B(D)} has at most {3} spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)

b) Prove that for any knot and any diagram {D}, {\Gamma_B(D)} has an odd number of spanning trees.

11. Let {X_1,X_2,..} be independent random variables with the same distribution, and let {S_n=X_1+X_2+...+X_n,\ n=1,2,...}. For what real numbers {c} is the following statement true:

\displaystyle P\left(\left| \frac{S_{2n}}{2n}- c \right| \leq \left| \frac{S_n}{n}-c\right| \right)\geq \frac{1}{2}\ \ ?

Miklos Schweitzer competition 2012 (many thanks to Eles Andras for the translation of the problems from Hungarian to English)

Partition of a set of sets

July 25, 2010 Leave a comment

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.
Miklos Schweitzer 2007

Least common multiple

June 21, 2010 Leave a comment

Given a positive integer n > 3, prove that the least common multiple of products of the form x_1x_2...x_k\ (k \geq 1) whose factors x_i are positive integers with x_1+x_2+...+x_n \leq n is less than n!.
Miklos Schweitzer 1951

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