CONO-SUR 2012

Around a circumference are written $2012$ number, each of with is equal to $1$ or $-1$ . If there are not $10$ consecutive numbers that sum $0$ , find all possible values of the sum of the $2012$ numbers.

In a square $ABCD$ , let $P$ be a point in the side $CD$ , different from $C$ and $D$ . In the triangle $ABP$ , the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$ . Show that $\angle ASB=90$ .

Show that there do not exist positive integers $a$ , $b$ , $c$ and $d$ , pairwise co-prime, such that $ab+cd$ , $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$ .

Find the biggest positive integer $n$ , lesser thar $2012$ , that has the following property: If $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ .

$A$ and $B$ play alternating turns on a $2012 \times 2013$ board with enough pieces of the following types:

Type $1$ : Piece like Type $2$ but with one square at the right of the bottom square. Type $2$ : Piece of $2$ consecutive squares, one over another. Type $3$ : Piece of $1$ square.

At his turn, $A$ must put a piece of the type $1$ on available squares of the board. $B$ , at his turn, must put exactly one piece of each type on available squares of the board. The player that cannot do more movements loses. If $A$ starts playing, decide who has a winning strategy.

Note: The pieces can be rotated but cannot overlap; they cannot be out of the board. The pieces of the types $1$ , $2$ and $3$ can be put on exactly $3$ , $2$ and $1$ squares of the board respectively.

Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$ . Let $M$ and $N$ , respectively, be variable points of the sides $AB$ and $AC$ , different from $A$ , such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$ . Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$ .