1995 IMC

Part of IMC

Subcontests

(12)

1

IMC 1995 Problem 12

(f_{n})_{n=1}^{\infty}

is a sequence of continuous functions on the interval

[0,1]

\int_{0}^{1}f_{m}(x)f_{n}(x) dx= \begin{cases} 1& \text{if}\;n=m\\ 0 & \text{if} \;n\ne m \end{cases}

\sup\{|f_{n}(x)|: x\in [0,1]\, \text{and}\, n=1,2,\dots\}< \infty

. Show that there exists no subsequence

(f_{n_{k}})

(f_{n})

\lim_{k\to \infty}f_{n_{k}}(x)

x\in [0,1]

1

IMC 1995 Problem 11

a) Prove that every function of the form

f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)

|a_{0}|<1

has positive as well as negative values in the period

[0,2\pi)

. b) Prove that the function

F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)

40

zeroes in the interval

(0,1000)

1

IMC 1995 Problem 10

a) Prove that for every

\epsilon>0

there is a positive integer

n

and real numbers

\lambda_{1},\dots,\lambda_{n}

\max_{x\in [-1,1]}|x-\sum_{k=1}^{n}\lambda_{k}x^{2k+1}|<\epsilon.

b) Prove that for every odd continuous function

f

[-1,1]

\epsilon>0

there is a positive integer

n

and real numbers

\mu_{1},\dots,\mu_{n}

\max_{x\in [-1,1]}|f(x)-\sum_{k=1}^{n}\mu_{k}x^{2k+1}|<\epsilon.

1

IMC 1995 Problem 9

Let all roots of an

n

-th degree polynomial

P(z)

with complex coefficients lie on the unit circle in the complex plane. Prove that all roots of the polynomial

2zP'(z)-nP(z)

lie on the same circle.

1

IMC 1995 Problem 8

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

be a sequence of positive real numbers such that

b_{0}=1

b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}

\sum_{n=1}^{\infty}b_{n}2^{n}.

1

IMC 1995 Problem 7

A

3\times 3

real matrix such that the vectors

Au

u

are orthogonal for every column vector

u\in \mathbb{R}^{3}

. Prove that: a)

A^{T}=-A

. b) there exists a vector

v \in \mathbb{R}^{3}

Au=v\times u

u\in \mathbb{R}^{3}

v \times u

denotes the vector product in

\mathbb{R}^{3}

1

IMC 1995 Problem 6

p>1

. Show that there exists a constant

K_{p} >0

such that for every

x,y\in \mathbb{R}

|x|^{p}+|y|^{p}=2

(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).

1

IMC 1995 Problem 5

A

B

n\times n

matrices. Assume there exist

n+1

different real numbers

t_{1},t_{2},\dots,t_{n+1}

such that the matrices

C_{i}=A+t_{i}B, \,\, i=1,2,\dots,n+1

are nilpotent. Show that both

A

B

1

IMC 1995 Problem 4

F:(1,\infty) \rightarrow \mathbb{R}

be the function defined by

F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}.

F

is injective and find the set of values of

F

1

IMC 1995 Problem 3

f

be twice continuously differentiable on

(0,\infty)

\lim_{x \to 0^{+}}f'(x)=-\infty

\lim_{x \to 0^{+}}f''(x)=\infty

\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.

1

IMC 1995 Problem 2

f

be a continuous function on

[0,1]

such that for every

x\in [0,1]

\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}

\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}

1

IMC 1995 Problem 1

X

be a invertible matrix with columns

X_{1},X_{2}...,X_{n}

Y

be a matrix with columns

X_{2},X_{3},...,X_{n},0

. Show that the matrices

A=YX^{-1}

B=X^{-1}Y

n-1

0

´s for eigenvalues.