MathDB

1

x_{n+1}=y_n +1/x_n, y_{n+1}=z_n +1/y_n, z_{n+1}=x_n +1/z_n

(x_n), (y_n), (z_n)

are given by

x_{n+1}=y_n +\frac{1}{x_n}

,

y_{n+1}=z_n +\frac{1}{y_n}

,

z_{n+1}=x_n +\frac{1}{z_n}

for

n \ge 0

where

x_0,y_0, z_0

are given positive numbers. Prove that these sequences are unbounded.

4

1

n cells occupied by 0 or 1 arranged into a circle

Let

n \ge 3

cells be arranged into a circle. Each cell can be occupied by

0

or

1

. The following operation is admissible: Choose any cell

C

occupied by a

1

, change it into a

0

and simultaneously reverse the entries in the two cells adjacent to

C

(so that

x,y

become

1 - x

,

1 - y

). Initially, there is a

1

in one cell and zeros elsewhere. For which values of

n

is it possible to obtain zeros in all cells in a finite number of admissible steps?

7

1

a^{2^n} - 1 has at least n + 1 distinct prime divisors

Let

a > 3

be an odd integer. Show that for every positive integer

n

the number

a^{2^n}- 1

has at least

n + 1

distinct prime divisors.

9

1

|f^n(x)-f^n(y)| < |x-y| for x, y\in [0,1], exists unique x_0 : f (x_0) = x_0

For a function

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

we define

f^1 = f

and

f^{n+1} (x) = f (f^n(x))

for

0 \le x \le 1

and

n \in N

. Given that there is a

n

such that

|f^n(x) - f^n(y)| < |x - y|

for all distinct

x, y \in [0,1]

, prove that there is a unique

x_0 \in [0,1]

such that

f (x_0) = x_0

.

8

1

partition plane to N regions by three bunches of parallel lines

The plane has been partitioned into

N

regions by three bunches of parallel lines. What is the least number of lines needed in order that

N > 1981

?

5

1

P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 with a_i rational has 1 real root

Let

P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4

be a polynomial with rational coefficients. Show that if

P(x)

has exactly one real root

\xi

, then

\xi

is a rational number.

2

1

a_{n+1} = a_n^2 + (a_n - 1)^2 , a_q - ap = a_m - a_k

The sequence

a_0, a_1, a_2, ...

is defined by

a_{n+1} = a^2_n + (a_n - 1)^2

for

n \ge 0

. Find all rational numbers

a_0

for which there exist four distinct indices

k, m, p, q

such that

a_q - a_p = a_m - a_k

.

1

Set and smalles integer

Find the smallest

n

for which we can find

15

distinct elements

a_{1},a_{2},...,a_{15}

of

\{16,17,...,n\}

such that

a_{k}

is a multiple of

k

.

3

1

Geometric inequality 1

Given is a triangle

ABC

, the inscribed circle

G

of which has radius

r

. Let

r_a

be the radius of the circle touching

AB

,

AC

and

G

. [This circle lies inside triangle

ABC

.] Define

r_b

and

r_c

similarly. Prove that

r_a + r_b + r_c \geq r

and find all cases in which equality occurs. Bosnia - Herzegovina Mathematical Olympiad 2002

1981 Austrian-Polish Competition

Subcontests

x_{n+1}=y_n +1/x_n, y_{n+1}=z_n +1/y_n, z_{n+1}=x_n +1/z_n

n cells occupied by 0 or 1 arranged into a circle

a^{2^n} - 1 has at least n + 1 distinct prime divisors

|f^n(x)-f^n(y)| < |x-y| for x, y\in [0,1], exists unique x_0 : f (x_0) = x_0

partition plane to N regions by three bunches of parallel lines

P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 with a_i rational has 1 real root

a_{n+1} = a_n^2 + (a_n - 1)^2 , a_q - ap = a_m - a_k

Set and smalles integer

Geometric inequality 1

1981 Austrian-Polish Competition

Subcontests

x_{n+1}=y_n +1/x_n, y_{n+1}=z_n +1/y_n, z_{n+1}=x_n +1/z_n

n cells occupied by 0 or 1 arranged into a circle

a^{2^n} - 1 has at least n + 1 distinct prime divisors

|f^n(x)-f^n(y)| &lt; |x-y| for x, y\in [0,1], exists unique x_0 : f (x_0) = x_0

partition plane to N regions by three bunches of parallel lines

P(x) = x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 with a_i rational has 1 real root

a_{n+1} = a_n^2 + (a_n - 1)^2 , a_q - ap = a_m - a_k

Set and smalles integer

Geometric inequality 1

|f^n(x)-f^n(y)| < |x-y| for x, y\in [0,1], exists unique x_0 : f (x_0) = x_0