MathDB

2

Functional equation with powers of 2

f:\mathbb{N}\rightarrow \mathbb{N}

such that for all

m,n\in\mathbb{N}

,

(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n.

Sum of B-entries equals the sum of corresponding A-entries

Given real numbers

x_i,y_i (i=1,2,\ldots ,n)

, let

A

be the

n\times n

matrix given by

a_{ij}=1

if

x_i\ge y_j

and

a_{ij}=0

otherwise. Suppose

B

is a

n\times n

matrix whose entries are

0

and

1

such that the sum of entries in any row or column of

B

equals the sum of entries in the corresponding row or column of

A

. Prove that

B=A

.

2

1

Infintiely many polygons coinciding with P_0

Let

\mathcal{P}_0=A_0A_1\ldots A_{n-1}

be a convex polygon such that

A_iA_{i+1}=2^{[i/2]}

for

i=0, 1,\ldots ,n-1

(where

A_n=A_0

). Define the sequence of polygons

\mathcal{P}_k=A_0^kA_1^k\ldots A_{n-1}^k

as follows:

A_i^1

is symmetric to

A_i

with respect to

A_0

,

A_i^2

is symmetric to

A_i^1

with respect to

A_1^1

,

A_i^3

is symmetric to

A_i^2

with respect to

A_2^2

and so on. Find the values of

n

for which infinitely many polygons

\mathcal{P}_k

coincide with

\mathcal{P}_0

.

1

2

Italian TST 2004 - Problem 1

At the vertices

A, B, C, D, E, F, G, H

of a cube,

2001, 2002, 2003, 2004, 2005, 2008, 2007

and

2006

stones respectively are placed. It is allowed to move a stone from a vertex to each of its three neighbours, or to move a stone to a vertex from each of its three neighbours. Which of the following arrangements of stones at

A, B, \ldots , H

can be obtained? (\text{a}) 2001, 2002, 2003, 2004, 2006, 2007, 2008, 2005; (\text{b}) 2002, 2003, 2004, 2001, 2006, 2005, 2008, 2007; (\text{c}) 2004, 2002, 2003, 2001, 2005, 2008, 2007, 2006.

The lines r,s,t and the two γ circles

Two circles

\gamma_1

and

\gamma_2

intersect at

A

and

B

. A line

r

through

B

meets

\gamma_1

at

C

and

\gamma_2

at

D

so that

B

is between

C

and

D

. Let

s

be the line parallel to

AD

which is tangent to

\gamma_1

at

E

, at the smaller distance from

AD

. Line

EA

meets

\gamma_2

in

F

. Let

t

be the tangent to

\gamma_2

at

F

.

(a)

Prove that

t

is parallel to

AC

.

(b)

Prove that the lines

r,s,t

are concurrent.

2004 Italy TST

Subcontests

Functional equation with powers of 2

Sum of B-entries equals the sum of corresponding A-entries

Infintiely many polygons coinciding with P_0

Italian TST 2004 - Problem 1

The lines r,s,t and the two γ circles