2003 IMC

Part of IMC

Subcontests

(6)

2

IMC 2003 Problem 5

g:[0,1]\rightarrow \mathbb{R}

be a continuous function and let

f_{n}:[0,1]\rightarrow \mathbb{R}

be a sequence of functions defined by

f_{0}(x)=g(x)

f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.

\lim_{n\to \infty}f_{n}(x)

x\in (0,1]

IMC 2003 Problem 11

a) Show that for each function

f:\mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{R}

, there exists a function

g:\mathbb{Q}\rightarrow \mathbb{R}

f(x,y) \leq g(x)+g(y)

x,y\in \mathbb{Q}

. b) Find a function

f:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}

, for which there is no function

g:\mathbb{Q}\rightarrow \mathbb{R}

f(x,y) \leq g(x)+g(y)

x,y\in \mathbb{R}

2

Idempotent

A\in\mathbb{R}^{n\times n}

3A^3=A^2+A+I

. Show that the sequence

A^k

converges to an idempotent matrix. (idempotent:

B^2=B

IMC 2003 Problem 9

A

be a closed subset of

\mathbb{R}^{n}

B

be the set of all those points

b \in \mathbb{R}^{n}

for which there exists exactly one point

a_{0}\in A

|a_{0}-b|= \inf_{a\in A}|a-b|

B

\mathbb{R}^{n}

; that is, the closure of

B

\mathbb{R}^{n}

2

51 elements of a field

a_1, a_2,...,a_{51}

be non-zero elements of a field of characteristic

p

. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence

b_1, ... , b_{51}

. If this new sequence is a permutation of the original one, find all possible values of

p

Limit of an integral

\lim_{x\rightarrow 0^+}\int^{2x}_x\frac{\sin^m(t)}{t^n}dt

m,n\in\mathbb{N}

2

Limit of a sequence (nice, easy)

a_1,a_2,...

be a sequenceof reals with

a_1=1

a_{n+1}>\frac32 a_n

n

\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}

exists. (finite or infinite) (b) Prove that for all

\alpha>1

there is a sequence

a_1,a_2,...

with the same properties such that

\lim_{n\rightarrow\infty}\frac{a_n}{\left(\frac32\right)^{n-1}}=\alpha

Commuting

A,B \in \mathbb{R}^{n\times n}

AB+B+A=0

AB=BA

2

IMC 2003 / A4

Determine the set of all pairs (a,b) of positive integers for which the set of positive integers can be decomposed into 2 sets A and B so that

a\cdot A=b\cdot B

IMC 2003 Problem 10

Find all the positive integers

n

for which there exists a Family

\mathcal{F}

of three-element subsets of

S=\{1,2,...,n\}

\text{(i) for any two different elements $a,b \in S$ there exists exactly one $A \in \mathcal{F}$ containing both $a$ and $b$;}

\text{(ii) if $a,b,c,x,y,z$ are elements of $S$ such that $\{a,b,x\},\{a,c,y\},\{b,c,z\} \in \mathcal{F}$, then $\{x,y,z\} \in \mathcal{F} $ }.

2

polynomial

p=\sum\limits_{k=0}^n a_kX^k\in R[X]

a polynomial such that all his roots lie in the half plane

\{z\in C| Re(z)<0 \}.

a_ka_{k+3}<a_{k+1}a_{k+2},

for every k=0,1,2...,n-3.

IMC 2003 Problem 12

(a_{n})

be the sequence defined by

a_{0}=1,a_{n+1}=\sum_{k=0}^{n}\dfrac{a_k}{n-k+2}

. Find the limit

\lim_{n \rightarrow \infty} \sum_{k=0}^{n}\dfrac{a_{k}}{2^{k}},