MathDB

1

Part of 2011 Pre-Preparation Course Examination

Problems(4)

Machine and Cards

Source: Iran pre-preparation course examination 2011- P1

2/25/2011
We have some cards that have the same look, but at the back of some of them is written 00 and for the others 11.(We can't see the back of a card so we can't know what's the number on it's back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have mm cards with 00 on their back and nn cards with 11 on their back, at least how many times we must use the machine to be sure that we get the number 11? (15 points)
combinatorics proposedcombinatorics
analytic function

Source:

2/26/2011
a) prove that the function ζ(s)=∑n=1∞1ns\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s} that is defined on the area Re(s)>1Re(s)>1, is an analytic function.
b) prove that the function ζ(s)−1s−1\zeta(s)-\frac{1}{s-1} can be spanned to an analytic function over C\mathbb C.
c) using the span of part b show that ζ(1−n)=−Bnn\zeta(1-n)=-\frac{B_n}{n} that BnB_n is the nnth bernoli number that is defined by generating function tet−1=∑n=0∞Bntnn!\frac{t}{e^t-1}=\sum_{n=0}^{\infty}B_n\frac{t^n}{n!}.
functiongeometryadvanced fieldsadvanced fields unsolved
invarient size on permutations of N

Source:

3/4/2011
suppose that SNS_{\mathbb N} is the set of all permutations of natural numbers. finite permutations are a subset of SNS_{\mathbb N} that behave like the identity permutation from somewhere. in other words bijective functions like π:N⟶N\pi: \mathbb N \longrightarrow \mathbb N that only for finite natural numbers ii, π(i)≠i\pi(i)\neq i. prove that we cannot put probability measure that is countably additive on ℘(SN)\wp(S_{\mathbb N}) (family of all the subsets of SNS_{\mathbb N}) that is invarient under finite permutations.
functionprobabilityprobability and stats
compressed set

Source:

2/27/2011
a) prove that for every compressed set KK in the space R3\mathbb R^3, the function f:R3⟶Rf:\mathbb R^3 \longrightarrow \mathbb R that f(p)=inf{∣p−k∣,k∈K}f(p)=inf\{|p-k|,k\in K\} is continuous. b) prove that we cannot cover the sphere S2⊆R3S^2\subseteq \mathbb R^3 with it's three closed sets, such that none of them contain two antipodal points.
functiongeometry3D geometryspheretopology