MathDB

2

Part of 2011 Pre-Preparation Course Examination

Problems(4)

Covering with dominos

Source: Iran pre-preparation course examination 2011- P2

2/25/2011
We say that a covering of a m×nm\times n rectangle with dominos has a wall if there exists a horizontal or vertical line that splits the rectangle into two smaller rectangles and doesn't cut any of the dominos. prove that if these three conditions are satisfied:
a) mnmn is an even number
b) m≥5m\ge 5 and n≥5n\ge 5
c) (m,n)≠(6,6)(m,n)\neq(6,6)
then we can cover the rectangle with dominos in such a way that we have no walls. (20 points)
geometryrectanglecombinatorics proposedcombinatorics
computing some values of zeta function

Source:

2/26/2011
by using the formula πcot(πz)=1z+∑n=1∞2zz2−n2\pi cot(\pi z)=\frac{1}{z}+\sum_{n=1}^{\infty}\frac{2z}{z^2-n^2} calculate values of ζ(2k)\zeta(2k) on terms of bernoli numbers and powers of π\pi.
functiongeometric seriesadvanced fieldsadvanced fields unsolved
inapproximation of real numbers

Source:

2/28/2011
prove that for almost every real number α∈[0,1]\alpha \in [0,1] there exists natural number nα∈Nn_{\alpha} \in \mathbb N such that the inequality ∣α−pq∣≤1qn|\alpha-\frac{p}{q}|\le \frac{1}{q^n} for natural n≥nαn\ge n_{\alpha} and rational pq\frac{p}{q} has no answers.
inequalitiesprobabilityprobability and stats
abelian space

Source:

2/27/2011
prove that π1(X,x0)\pi_1 (X,x_0) is not abelian. XX is like an eight (8)(8) figure. comments: eight figure is the union of two circles that have one point x0x_0 in common. we call a group GG abelian if: ∀a,b∈G:ab=ba\forall a,b \in G:ab=ba.
abstract algebratopology