RIOPLATENSE 2011

Given a positive integer $n$ , an operation consists of replacing $n$ with either $2n-1$ , $3n-2$ or $5n-4$ . A number $b$ is said to be a follower of number $a$ if $b$ can be obtained from $a$ using this operation multiple times. Find all positive integers $a < 2011$ that have a common follower with $2011$ .

Let $ABC$ an acute triangle and $H$ its orthocenter. Let $E$ and $F$ be the intersection of lines $BH$ and $CH$ with $AC$ and $AB$ respectively, and let $D$ be the intersection of lines $EF$ and $BC$ . Let $\Gamma_1$ be the circumcircle of $AEF$ , and $\Gamma_2$ the circumcircle of $BHC$ . The line $AD$ intersects $\Gamma_1$ at point $I \neq A$ . Let $J$ be the feet of the internal bisector of $\angle{BHC}$ and $M$ the midpoint of the arc $\stackrel{\frown}{BC}$ from $\Gamma_2$ that contains the point $H$ . The line $MJ$ intersects $\Gamma_2$ at point $N \neq M$ . Show that the triangles $EIF$ and $CNB$ are similar.

Let $M$ be a map made of several cities linked to each other by flights. We say that there is a route between two cities if there is a nonstop flight linking these two cities. For each city to the $M$ denote by $M_a$ the map formed by the cities that have a route to and routes linking these cities together ( to not part of $M_a$ ). The cities of $M_a$ are divided into two sets so that the number of routes linking cities of different sets is maximum; we call this number the cut of $M_a$ . Suppose that for every cut of $M_a$ , it is strictly less than two thirds of the number of routes $M_a$ . Show that for any coloring of the $M$ routes with two colors there are three cities of $M$ joined by three routes of the same color.

We consider $\Gamma_1$ and $\Gamma_2$ two circles that intersect at points $P$ and $Q$ . Let $A , B$ and $C$ be points on the circle $\Gamma_1$ and $D , E$ and $F$ points on the circle $\Gamma_2$ so that the lines $A E$ and $B D$ intersect at $P$ and the lines $A F$ and $C D$ intersect at $Q$ . Denote $M$ and $N$ the intersections of lines $A B$ and $D E$ and of lines $A C$ and $D F$ , respectively. Show that $A M D N$ is a parallelogram.

A form is the union of squared rectangles whose bases are consecutive unitary segments in a horizontal line that leaves all the rectangles on the same side, and whose heights $m_1, ... , m_n$ satisying $m_1\ge ... \ge m_n$ . An angle in a form consists of a box $v$ and of all the boxes to the right of $v$ and all the boxes above $v$ . The size of a form of an angle is the number of boxes it contains. Find the maximum number of angles of size $11$ in a form of size $400$ .

source

Let $d(n)$ be the sum of positive integers divisors of number $n$ and $\phi(n)$ the quantity of integers in the interval $[0,n]$ such that these integers are coprime with $n$ . For instance $d(6)=12$ and $\phi(7)=6$ . Determine if the set of the integers $n$ such that, $d(n)\cdot \phi (n)$ is a perfect square, is finite or infinite set.