AUSTRIAN-POLISH 2004

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:

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

Solve the following system of equations in $\mathbb{R}$ where all square roots are non-negative:

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

Determine all $n \in \mathbb{N}$ for which $n^{10} + n^5 + 1$ is prime.

Determine all $n$ for which the system with of equations can be solved in $\mathbb{R}$ :

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

and

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

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: $a_j - a_i \neq a_k - a_j$ .

Determine all functions $f:\mathbb{Z}^+\to \mathbb{Z}$ which satisfy the following condition for all pairs $(x,y)$ of relatively prime positive integers: $$f(x+y) = f(x+1) + f(y+1).$$

a.) Prove that for $n = 4$ or $n \geq 6$ each triangle $ABC$ can be decomposed in $n$ similar (not necessarily congruent) triangles.

b.) Show: An equilateral triangle can neither be composed in 3 nor 5 triangles.

c.) Is there a triangle $ABC$ 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.

Given are the 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 $a_0 = 26, b_0 = 6, c_0 = 2004$ .

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 $|M(P_n)| = m_n.$