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$.

Click for solution This is easier than I had expected. Let $O(XYZ)$ be the circumcenter of $XYZ$, and $H(XYZ)$ the orthocenter of $XYZ$. By simple angle chasing we show that $O=O(A_2B_2C_2)\Leftrightarrow OA_2\perp BC (\mbox{ and } OB_2\perp AC,\ OC_2\perp AB)$. At the same time, the last condition I wrote is equivalent to quadrilateral $BCC_1B_1$ being cyclic (because the perpendicular bisectors of three of its sides concur), which is equivalent to $B_1C_1\perp AI$ (angle chase again), and doing this for all three quadrilaterals we get to the conclusion. Sorry I didn't write it in more details..

For any positive integer $n$ the sum $\displaystyle 1+\frac 12+ \cdots + \frac 1n$ is written in the form $\displaystyle \frac{P(n)}{Q(n)}$, 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 $\displaystyle \frac 2{5}n$ 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$.