OMM 2005

Let $O$ be the center of the circumcircle of an acute triangle $ABC$ , let $P$ be any point inside the segment $BC$ . Suppose the circumcircle of triangle $BPO$ intersects the segment $AB$ at point $R$ and the circumcircle of triangle $COP$ intersects $CA$ at point $Q$ . (i) Consider the triangle $PQR$ , show that it is similar to triangle $ABC$ and that $O$ is its orthocenter. (ii) Show that the circumcircles of triangles $BPO$ , $COP$ , $PQR$ have the same radius.

Given several matrices of the same size. Given a positive integer $N$ , let's say that a matrix is $N$ -balanced if the entries of the matrix are integers and the difference between any two adjacent entries of the matrix is less than or equal to $N$ . (i) Show that every $2N$ -balanced matrix can be written as a sum of two $N$ -balanced matrices. (ii) Show that every $3N$ -balanced matrix can be written as a sum of three $N$ -balanced matrices.

Already the complete problem:

Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers:

$\rightarrow \frac{a + xy}{b}$ , $\frac{a + xy^2}{b^2}$ , $\frac{a + xy^3}{b^3}$ , $\ldots$ , $\frac{a + xy^n}{b^n}$ , $\ldots$

One idea?

:arrow: View all the problems from XIX Mexican Mathematical Olympiad

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$ . Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

Let $N$ be an integer greater than $1$ . A deck has $N^3$ cards, each card has one of $N$ colors, has one of $N$ figures and has one of $N$ numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?

Let $ABC$ be a triangle and $AD$ be the angle bisector of $<BAC$ , with $D$ on $BC$ . Let $E$ be a point on segment $BC$ such that $BD = EC$ . Through $E$ draw $l$ a parallel line to $AD$ and let $P$ be a point in $l$ inside the triangle. Let $G$ be the point where $BP$ intersects $AC$ and $F$ be the point where $CP$ intersects $AB$ . Show $BF = CG$ .