RIOPLATENSE 1998

Consider an arc $AB$ of a circle $C$ and a point $P$ variable in that arc $AB$ . Let $D$ be the midpoint of the arc $AP$ that doeas not contain $B$ and let $E$ be the midpoint of the arc $BP$ that does not contain $A$ . Let $C_1$ be the circle with center $D$ passing through $A$ and $C_2$ be the circle with center $E$ passing through $B.$ Prove that the line that contains the intersection points of $C_1$ and $C_2$ passes through a fixed point.

Given an integer $n > 2$ , consider all sequences $x_1,x_2,...,x_n$ of nonnegative real numbers such that $$x_1+ 2x_2 + ... + nx_n = 1.$$ Find the maximum value and the minimum value of $x_1^2+x_2^2+...+x_n^2$ and determine all the sequences $x_1,x_2,...,x_n$ for which these values are obtained.

Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$ , there is a subset $B$ of $X$ , with the following property: For each element $e$ of $X$ , $e$ divides an odd number of elements of $B$ , if and only if $e$ is an element of $A$ .

Let $M$ be a subset of $\{1,2,..., 1998\}$ with $1000$ elements. Prove that it is always possible to find two elements $a$ and $b$ in $M$ , not necessarily distinct, such that $a + b$ is a power of $2$ .

We say that $M$ is the midpoint of the open polygonal $XYZ$ , formed by the segments $XY, YZ$ , if $M$ belongs to the polygonal and divides its length by half. Let $ABC$ be a acute triangle with orthocenter $H$ . Let $A_1, B_1,C_1,A_2, B_2,C_2$ be the midpoints of the open polygonal $CAB, ABC, BCA, BHC, CHA, AHB$ , respectively. Show that the lines $A_1 A_2, B_1B_2$ and $C_1C_2$ are concurrent.

Let $k$ be a fixed positive integer. For each $n = 1, 2,...,$ we will call configuration of order $n$ any set of $kn$ points of the plane, which does not contain $3$ collinear, colored with $k$ given colors, so that there are $n$ points of each color. Determine all positive integers $n$ with the following property: in each configuration of order $n$ , it is possible to select three points of each color, such that the $k$ triangles with vertices of the same color that are determined are disjoint in pairs.