BG-MO 2004

Let $I$ be the incenter of triangle $ABC$ , and let $A_1$ , $B_1$ , $C_1$ be arbitrary points on the segments $(AI)$ , $(BI)$ , $(CI)$ , respectively. The perpendicular bisectors of $AA_1$ , $BB_1$ , $CC_1$ intersect each other at $A_2$ , $B_2$ , and $C_2$ . Prove that the circumcenter of the triangle $A_2B_2C_2$ coincides with the circumcenter of the triangle $ABC$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$ .

For any positive integer $n$ the sum $1+\frac 12+ \cdots + \frac 1n$ is written in the form $\frac{P(n)}{Q(n)}$ , where $P(n)$ and $Q(n)$ are relatively prime.

a) Prove that $P(67)$ is not divisible by 3;

b) Find all possible $n$ , for which $P(n)$ is divisible by 3.

A group consist of n tourists. Among every 3 of them there are 2 which are not familiar. For every partition of the tourists in 2 buses you can find 2 tourists that are in the same bus and are familiar with each other. Prove that is a tourist familiar to at most $\frac 2{5}n$ tourists.

In a word formed with the letters $a,b$ we can change some blocks: $aba$ in $b$ and back, $bba$ in $a$ and backwards. If the initial word is $aaa\ldots ab$ where $a$ appears 2003 times can we reach the word $baaa\ldots a$ , where $a$ appears 2003 times.

Let $a,b,c,d$ be positive integers such that the number of pairs $(x,y) \in (0,1)^2$ such that both $ax+by$ and $cx+dy$ are integers is equal with 2004. If $\gcd (a,c)=6$ find $\gcd (b,d)$ .

Let $p$ be a prime number and let $0\leq a_{1}< a_{2}<\cdots < a_{m}< p$ and $0\leq b_{1}< b_{2}<\cdots < b_{n}< p$ be arbitrary integers. Let $k$ be the number of distinct residues modulo $p$ that $a_{i}+b_{j}$ give when $i$ runs from 1 to $m$ , and $j$ from 1 to $n$ . Prove that

a) if $m+n > p$ then $k = p$ ;

b) if $m+n\leq p$ then $k\geq m+n-1$ .