OMM 2006

Let $ab$ be a two digit number. A positive integer $n$ is a relative of $ab$ if:

• The units digit of $n$ is $b$ .
• The remaining digits of $n$ are nonzero and add up to $a$ .

Find all two digit numbers which divide all of their relatives.

Let $ABC$ be a right triangle with a right angle at $A$ , such that $AB < AC$ . Let $M$ be the midpoint of $BC$ and $D$ the intersection of $AC$ with the perpendicular on $BC$ passing through $M$ . Let $E$ be the intersection of the parallel to $AC$ that passes through $M$ , with the perpendicular on $BD$ passing through $B$ . Show that the triangles $AEM$ and $MCA$ are similar if and only if $\angle ABC = 60^o$ .

Let $n$ be an integer greater than $1$ . In how many ways can we fill all the numbers $1, 2,..., 2n$ in the boxes of a grid of $2\times n$ , one in each box, so that any two consecutive numbers are they in squares that share one side of the grid?

For which positive integers $n$ can be covered a ladder like that of the figure (but with $n$ steps instead of $4$ ) with $n$ squares of integer sides, not necessarily the same size, without these squares overlapping and without standing out from the edge of the figure ?

Attachments:

Let $ABC$ be an acute triangle , with altitudes $AD,BE$ and $CF$ . Circle of diameter $AD$ intersects the sides $AB,AC$ in $M,N$ respevtively. Let $P,Q$ be the intersection points of $AD$ with $EF$ and $MN$ respectively. Show that $Q$ is the midpoint of $PD$ .

Let n be the sum of the digits in a natural number A. The number A it's said to be "surtido" if every number 1,2,3,4....,n can be expressed as a sum of digits in A.

a)Prove that, if 1,2,3,4,5,6,7,8 are sums of digits in A, then A is "Surtido" b)If 1,2,3,4,5,6,7 are sums of digits in A, does it follow that A is "Surtido"?