MathDB

1

Some average sequences

(..., a_{-2}, a_{-1}, a_0, a_1, a_2, ...); (..., b_{-2}, b_{-1}, b_0, b_1, b_2, ...); (..., c_{-2}, c_{-1}, c_0, c_1, c_2, ...)

of positive real numbers. For each integer

n

the following inequalities hold:

a_n \geq \frac{1}{2} (b_{n+1} + c_{n-1})

b_n \geq \frac{1}{2} (c_{n+1} + a_{n-1})

c_n \geq \frac{1}{2} (a_{n+1} + b_{n-1})

Determine

a_{2005}

,

b_{2005}

,

c_{2005}

, if

a_0 = 26, b_0 = 6, c_0 = 2004

.

7

1

Functional equation on relative primes

Determine all functions

f:\mathbb{Z}^+\to \mathbb{Z}

which satisfy the following condition for all pairs

(x,y)

of relatively prime positive integers:

f(x+y) = f(x+1) + f(y+1).

6

1

Permutation of 1,2,...,n with no arithmetic subsequence

For

n=2^m

(m is a positive integer) consider the set

M(n) = \{ 1,2,...,n\}

of natural numbers. Prove that there exists an order

a_1, a_2, ..., a_n

of the elements of M(n), so that for all

1\leq i < j < k \leq n

holds:

a_j - a_i \neq a_k - a_j

.

1

Digit sum

Let

S(n)

be the sum of digits for any positive integer n (in decimal notation). Let

N=\displaystyle\sum_{k=10^{2003}}^{10{^{2004}-1}} S(k)

. Determine

S(N)

.

8

1

Austria-Poland triangle decomposition

a.) Prove that for

n = 4

or

n \geq 6

each triangle

ABC

can be decomposed in

n

ABC

which can be decomposed in 3 and 5 triangles, analogously to a.). Either give an example or prove that there is not such a triangle.

3

1

Austria-Poland 2004 system of equations

Solve the following system of equations in

\mathbb{R}

where all square roots are non-negative:

\begin{matrix} a - \sqrt{1-b^2} + \sqrt{1-c^2} = d \\ b - \sqrt{1-c^2} + \sqrt{1-d^2} = a \\ c - \sqrt{1-d^2} + \sqrt{1-a^2} = b \\ d - \sqrt{1-a^2} + \sqrt{1-b^2} = c \\ \end{matrix}

2

1

Austra-Poland triangle inequality

In a triangle

ABC

let

D

be the intersection of the angle bisector of

\gamma

, angle at

C

, with the side

AB.

And let

F

be the area of the triangle

ABC.

Prove the following inequality:

2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.

10

1

Austria-Poland 2004 polynomial

For each polynomial

Q(x)

let

M(Q)

be the set of non-negative integers

x

with

0 < Q(x) < 2004.

We consider polynomials

P_n(x)

of the form

P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1

with coefficients

a_i \in \{ \pm1\}

for

i = 1, 2, \ldots, n-1.

For each

n = 3^k, k > 0

determine: a.)

m_n

which represents the maximum of elements in

M(P_n)

for all such polynomials

P_n(x)

b.) all polynomials

P_n(x)

for which

|M(P_n)| = m_n.

4

1

well-known problem from Austria-Poland: n^{10} + n^5 + 1

Determine all

n \in \mathbb{N}

for which

n^{10} + n^5 + 1

is prime.

5

1

Austria-Poland where the sum is 27 and the product 24

Determine all

n

for which the system with of equations can be solved in

\mathbb{R}

:

\sum^{n}_{k=1} x_k = 27

and

\prod^{n}_{k=1} x_k = \left( \frac{3}{2} \right)^{24}.

2004 Austrian-Polish Competition

Subcontests

Some average sequences

Functional equation on relative primes

Permutation of 1,2,...,n with no arithmetic subsequence

Digit sum

Austria-Poland triangle decomposition

Austria-Poland 2004 system of equations

Austra-Poland triangle inequality

Austria-Poland 2004 polynomial

well-known problem from Austria-Poland: n^{10} + n^5 + 1

Austria-Poland where the sum is 27 and the product 24