AUSTRIAN-POLISH 1981

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 $k$ .

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 $a_q - a_p = a_m - a_k$ .

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

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?

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.

The sequences $(x_n), (y_n), (z_n)$ 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.

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.

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 $N > 1981$ ?

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 $f (x_0) = x_0$ .