OME 1989

An exam at a university consists of one question randomly selected from the $n$ possible questions. A student knows only one question, but he can take the exam $n$ times. Express as a function of $n$ the probability $p_n$ that the student will pass the exam. Does $p_n$ increase or decrease as $n$ increases? Compute $lim_{n\to \infty}p_n$ . What is the largest lower bound of the probabilities $p_n$ ?

Points $A' ,B' ,C'$ on the respective sides $BC,CA,AB$ of triangle $ABC$ satisfy $\frac{AC' }{AB} = \frac{BA' }{BC} = \frac{CB' }{CA} = k$ . The lines $AA' ,BB' ,CC'$ form a triangle $A_1B_1C_1$ (possibly degenerate). Given $k$ and the area $S$ of $\triangle ABC$ , compute the area of $\triangle A_1B_1C_1$ .

Show that the number $1989$ as well as each of its powers $1989^n$ ( $n \in N$ ), can be expressed as a sum of two positive squares in at least two ways.

Consider the set $D$ of all complex numbers of the form $a+b\sqrt{-13}$ with $a,b \in Z$ . The number $14 = 14+0\sqrt{-13}$ can be written as a product of two elements of $D$ : $14 = 2 \cdot 7$ . Find all possible ways to express $14$ as a product of two elements of $D$ .

Prove that among any seven real numbers there exist two, $a$ and $b$ , such that $\sqrt3|a-b|\le |1+ab|$ . Give an example of six real numbers not having this property.