AUSTRIAN-POLISH 2001

Determine the number of positive integers $a$ , so that there exist nonnegative integers $x_0,x_1,\ldots,x_{2001}$ which satisfy the equation $$a^{x_0} = \sum_{i=1}^{2001} a^{x_i}$$

Let $n$ be a positive integer greater than $2$ . Solve in nonnegative real numbers the following system of equations $$x_{k}+x_{k+1}=x_{k+2}^{2} , k=1,2,\cdots,n$$ where $x_{n+1}=x_{1}$ and $x_{n+2}=x_{2}$ .

Let $a,b,c$ be sides of a triangle. Prove that $$2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3$$

Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$ , then the following inequality holds $$S \leq \frac{1}{2}(ac+bd).$$ For which quadrangles does the inequality become equality?

The fields of the $8\times 8$ chessboard are numbered from $1$ to $64$ in the following manner: For $i=1,2,\cdots,63$ the field numbered by $i+1$ can be reached from the field numbered by $i$ by one move of the knight. Let us choose positive real numbers $x_{1},x_{2},\cdots,x_{64}$ . For each white field numbered by $i$ define the number $y_{i}=1+x_{i}^{2}-\sqrt[3]{x_{i-1}^{2}x_{i+1}}$ and for each black field numbered by $j$ define the number $y_{j}=1+x_{j}^{2}-\sqrt[3]{x_{j-1}x_{j+1}^{2}}$ where $x_{0}=x_{64}$ and $x_{1}=x_{65}$ . Prove that $$\sum_{i=1}^{64}y_{i}\geq 48$$

Consider the set $A$ containing all positive integers whose decimal expansion contains no $0$ , and whose sum $S(N)$ of the digits divides $N$ .

(a) Prove that there exist infinitely many elements in $A$ whose decimal expansion contains each digit the same number of times as each other digit.

(b) Explain that for each positive integer $k$ there exist an element in $A$ having exactly $k$ digits.

The prism with the regular octagonal base and with all edges of the length equal to $1$ is given. The points $M_{1},M_{2},\cdots,M_{10}$ are the midpoints of all the faces of the prism. For the point $P$ from the inside of the prism denote by $P_{i}$ the intersection point (not equal to $M_{i}$ ) of the line $M_{i}P$ with the surface of the prism. Assume that the point $P$ is so chosen that all associated with $P$ points $P_{i}$ do not belong to any edge of the prism and on each face lies exactly one point $P_{i}$ . Prove that $$\sum_{i=1}^{10}\frac{M_{i}P}{M_{i}P_{i}}=5$$

Let $A$ be a set with $2n$ elements, and let $A_1, A_2...,A_m$ be subsets of $A$ e ach one with n elements. Find the greatest possible m, such that it is possible to select these $m$ subsets in such a way that the intersection of any 3 of them has at most one element.

The sequence $a_{1},a_{2},\cdots,a_{2010}$ has the following properties:

(1) each sum of the 20 successive values of the sequence is nonnegative,

(2) $|a_{i}a_{i+1}| \leq 1$ for $i=1,2,\cdots,2009$ .

Determine the maximal value of the expression $\sum_{i=1}^{2010}a_{i}$ .