MathDB

1

f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0, nxn functional system

n \ge 2

, find all sustems of

n

functions

f_1,..., f_n : R \to R

such that for all

x,y \in R

\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}

6

1

x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 for n = 1,2,..., 1999, x_0 = x_{1999}.

Solve in the nonnegative real numbers the system of equations

\begin{cases} x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 \,\,\,\, for \,\,\,\, n = 1,2,..., 1999 \\\ x_0 = x_{1999} \end{cases}

9

1

lattice point game

A point in the cartesian plane with integer coordinates is called a lattice point. Consider the following one player game. A finite set of selected lattice points and finite set of selected segments is called a position in this game if the following hold: (i) The endpoints of each selected segment are lattice points; (ii) Each selected segment is parallel to a coordinate axis or to one of the lines

y = \pm x

, (iii) Each selected segment contains exactly five lattice points, all of which are selected, (iv) Every two selected segments have at most one common point. A move in this game consists of selecting a lattice point and a segment such that the new set of selected lattice points and segments is a position. Prove or disprove that there exists an initial position such that the game can have infinitely many moves.

7

1

x^{x+y} =y^{y-x} diophantine

Find all pairs

(x,y)

of positive integers such that

x^{x+y} =y^{y-x}

.

5

1

a_{n+1} = a_n^3 + 1999, exists at most 1 perfect square

A sequence of integers

(a_n)

satisfies

a_{n+1} = a_n^3 + 1999

for

n = 1,2,....

Prove that there exists at most one

n

for which

a_n

is a perfect square.

1

each element of M belongs to 0, 3, or 6 of the subsets $A_1,...,A_6

Find the number of

6

-tuples

(A_1,A_2,...,A_6)

of subsets of

M = \{1,..., n\}

(not necessarily different) such that each element of

M

belongs to zero, three, or six of the subsets

A_1,...,A_6

.

4

1

find a point such thath 3 triangles have the same area

Three lines

k, l, m

are drawn through a point

P

inside a triangle

ABC

such that

k

meets

AB

at

A_1

and

AC

at

A_2 \ne A_1

and

PA_1 = PA_2

,

l

meets

BC

at

B_1

and

BA

at

B_2 \ne B_1

and

PB_1 = PB_2

,

m

meets

CA

at

C_1

and

CB

at

C_2\ne C_1

and

PC_1=PC_2

. Prove that the lines

k,l,m

are uniquely determined by these conditions. Find point

P

for which the triangles

AA_1A_2, BB_1B_2, CC_1C_2

have the same area and show that this point is unique.

8

1

midpoint, related to perpendiculars, circumcircle - Austrian-Polish 1999

Let

P,Q,R

be points on the same side of a line

g

in the plane. Let

M

and

N

be the feet of the perpendiculars from

P

and

Q

to

g

respectively. Point

S

lies between the lines

PM

and

QN

and satisfies and satisfies

PM = PS

and

QN = QS

. The perpendicular bisectors of

SM

and

SN

meet in a point

R

. If the line

RS

intersects the circumcircle of triangle

PQR

again at

T

, prove that

S

is the midpoint of

RT

.

2

1

Find the best constants

Find the best possible

k,k'

such that

k<\frac{v}{v+w}+\frac{w}{w+x}+\frac{x}{x+y}+\frac{y}{y+z}+\frac{z}{z+v}<k'

for all positive reals

v,w,x,y,z

.

1999 Austrian-Polish Competition

Subcontests

f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0, nxn functional system

x_n^2 + x_nx_{n-1} + x_{n-1}^4 = 1 for n = 1,2,..., 1999, x_0 = x_{1999}.

lattice point game

x^{x+y} =y^{y-x} diophantine

a_{n+1} = a_n^3 + 1999, exists at most 1 perfect square

each element of M belongs to 0, 3, or 6 of the subsets $A_1,...,A_6

find a point such thath 3 triangles have the same area

midpoint, related to perpendiculars, circumcircle - Austrian-Polish 1999

Find the best constants