2010 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

circles inside square, perimeter sum

Inside a square consider circles such that the sum of their circumferences is twice the perimeter of the square.
a) Find the minimum number of circles having this property. b) Prove that there exist infinitely many lines which intersect at least 3 of these circles.

1

egyptian fractions

A

B

n\times n

matrices with integer entries, and

\det B\ne0

. Prove that there exists

m\in\mathbb N

such that the product

AB^{-1}

can be represented as

AB^{-1}=\sum_{k=1}^mN_k^{-1},

N_k

n\times n

matrices with integer entries for all

k=1,\ldots,m

N_i\ne N_j

i\ne j

1

M2(R), existence of matrices satisfying conditions

\mathcal M_2(\mathbb R)

2\times2

matrices with real entries. Prove that:
a) for every

A\in\mathcal M_2(\mathbb R)

B,C\in\mathcal M_2(\mathbb R)

A=B^2+C^2

; b) there do not exist

B,C\in\mathcal M_2(\mathbb R)

\begin{pmatrix}0&1\\1&0\end{pmatrix}=B^2+C^2

BC=CB

1

function sequence, n-repeated integral

f_0:[0,1]\to\mathbb R

be a continuous function. Define the sequence of functions

f_n:[0,1]\to\mathbb R

f_n(x)=\int^x_0f_{n-1}(t)dt

for all integers

n\ge1

.
a) Prove that the series

\sum_{n=1}^\infty f_n(x)

is convergent for every

x\in[0,1]

. b) Find an explicit formula for the sum of the series

\sum_{n=1}^\infty f_n(x),x\in[0,1]