CONO-SUR 2016

Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$ . Let $\overline{abcd}$ be an special number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$ . For example 2016 is special. Find all the $\overline{abcd}$ special numbers.Note: If $\overline{abcd}=0206$ , then $ab=02$ and $cd=06$ .

For every $k= 1,2, \ldots$ let $s_k$ be the number of pairs $(x,y)$ satisfying the equation $kx + (k+1)y = 1001 - k$ with $x$ , $y$ non-negative integers. Find $s_1 + s_2 + \cdots + s_{200}$ .

There are $2016$ positions marked around a circle, with a token on one of them. A legitimate move is to move the token either 1 position or 4 positions from its location, clockwise. The restriction is that the token can not occupy the same position more than once. Players $A$ and $B$ take turns making moves. Player $A$ has the first move. The first player who cannot make a legitimate move loses. Determine which of the two players has a winning strategy.

Let $S(n)$ be the sum of the digits of the positive integer $n$ . Find all $n$ such that $S(n)(S(n)-1)=n-1$ .

Let $ABC$ be a triangle inscribed on a circle with center $O$ . Let $D$ and $E$ be points on the sides $AB$ and $BC$ ,respectively, such that $AD = DE = EC$ . Let $X$ be the intersection of the angle bisectors of $\angle ADE$ and $\angle DEC$ . If $X \neq O$ , show that, the lines $OX$ and $DE$ are perpendicular.

We say that three different integers are friendly if one of them divides the product of the other two. Let $n$ be a positive integer.

a) Show that, between $n^2$ and $n^2+n$ , exclusive, does not exist any triplet of friendly numbers.

b) Determine if for each $n$ exists a triplet of friendly numbers between $n^2$ and $n^2+n+3\sqrt{n}$ , exclusive.