Miklos Schweitzer 2012 Problems

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)

  1. January 7, 2013 at 5:18 pm

    Great! Thank you.

  1. January 8, 2013 at 6:20 pm

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