1996 IMC

Part of IMC

Subcontests

(12)

1

IMC 1996 Problem 1

A=(a_{ij})\in M_{(n+1)\times (n+1)}(\mathbb{R})

a_{ij}=a+|i-j|d

a

d

are fixed real numbers. Calculate

\det(A)

1

IMC 1996 Problem 6

Upper content of a subset

E

\mathbb{R}^{2}

\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}

\inf

is taken over all finite families of sets

E_{1},\dots,E_{n}

n\in \mathbb{N}

\mathbb{R}^{2}

E\subset \bigcup_{i=1}^{n}E_{i}

. Lower content of

E

\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}

. Prove that i)

\mathcal{C}(L)=\text{length}(L)

L

is a closed line segment; ii)

\mathcal{C}(E) \geq \mathcal{K}(E)

; iii) the equality in ii) is not always true even if

E

1

IMC 1996 Problem 5

a,b

be real numbers such that

b\leq 0

1+ax+bx^{2} \geq 0

x\in [0,1]

\lim_{n\to \infty} n \int_{0}^{1}(1+ax+bx^{2})^{n}dx= \begin{cases} -\frac{1}{a} &\text{if}\; a<0,\\ \infty & \text{if}\; a \geq 0. \end{cases}

f:[0,1]\rightarrow[0,\infty)

be a function with a continuous second derivative and let

f''(x)\leq0

x\in [0,1]

L=\lim_{n\to \infty} n \int_{0}^{1}(f(x))^{n}dx

0<L<\infty

f'

has a constant sign and

\min_{x\in [0,1]}|f'(x)|=L^{-1}

1

IMC 1996 Problem 4

a_{1}=1

a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}

n\geq 2

\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}

\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}

1

IMC 1996 Problem 3

The linear operator

A

on a finite-dimensional vector space

V

is called an involution if

A^{2}=I

I

is the identity operator. Let

\dim V=n

. i) Prove that for every involution

A

V

, there exists a basis of

V

consisting of eigenvectors of

A

. ii) Find the maximal number of distinct pairwise commuting involutions on

V

1

IMC 1996 Problem 2

Evaluate the definite integral

\int_{-\pi}^{\pi}\frac{\sin nx}{(1+2^{x})\sin x} dx,

n

is a natural number.

1

IMC 1996 Problem 12

i) Prove that for every sequence

(a_{n})_{n\in \mathbb{N}}

a_{n}>0

n \in \mathbb{N}

\sum_{n=1}^{\infty}a_{n}<\infty

\sum_{n=1}^{\infty}(a_{1}a_{2} \cdots a_{n})^{\frac{1}{n}}< e\sum_{n=1}^{\infty}a_{n}.

ii) Prove that for every

\epsilon>0

there exists a sequence

(b_{n})_{n\in \mathbb{N}}

b_{n}>0

n \in \mathbb{N}

\sum_{n=1}^{\infty}b_{n}<\infty

\sum_{n=1}^{\infty}(b_{1}b_{2} \cdots b_{n})^{\frac{1}{n}}> (e-\epsilon)\sum_{n=1}^{\infty}b_{n}.

1

IMC 1996 Problem 11

\lim_{x\to \infty}\,\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}=\frac{1}{2}

. ii) Prove that there is a positive constant

c

such that for every

x\in [1,\infty)

\left|\sum_{n=1}^{\infty} \frac{nx}{(n^{2}+x)^{2}}-\frac{1}{2} \right| \leq \frac{c}{x}

1

IMC 1996 Problem 10

B

be a bounded closed convex symmetric (with respect to the origin) set in

\mathbb{R}^{2}

\Gamma

B

have the property that the ellipse of maximal area contained in

B

D

1

centered at the origin with boundary

C

A \cap \Gamma \ne \emptyset

A

C

l(A)\geq \frac{\pi}{2}

1

IMC 1996 Problem 9

G

be the subgroup of

GL_{2}(\mathbb{R})

A

B

A=\begin{pmatrix} 2 &0\\ 0&1 \end{pmatrix},\; B=\begin{pmatrix} 1 &1\\ 0&1 \end{pmatrix}

H

consist of the matrices

\begin{pmatrix} a_{11} &a_{12}\\ a_{21}& a_{22} \end{pmatrix}

G

a_{11}=a_{22}=1

H

is an abelian subgroup of

G

H

is not finitely generated.

1

IMC 1996 Problem 8

\theta

be a positive real number. Show that if

k\in \mathbb{N}

\cosh k \theta

\cosh(k+1) \theta

are rational, then so is

\cosh \theta

1

IMC 1996 Problem 7

f:[0,1]\rightarrow[0,1]

is a continuous function, then the sequence of iterates

x_{n+1}=f(x_{n})

converges if and only if

\lim_{n\to \infty}(x_{n+1}-x_{n})=0