OMM 2002

The numbers $1$ to $1024$ are written one per square on a $32 \times 32$ board, so that the first row is $1, 2, ... , 32$ , the second row is $33, 34, ... , 64$ and so on. Then the board is divided into four $16 \times 16$ boards and the position of these boards is moved round clockwise, so that $AB$ goes to $DA$ $DC \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, CB$ then each of the $16 \times 16$ boards is divided into four equal $8 \times 8$ parts and each of these is moved around in the same way (within the $16 \times 16$ board). Then each of the $8 \times 8$ boards is divided into four $4 \times 4$ parts and these are moved around, then each $4 \times 4$ board is divided into $2 \times 2$ parts which are moved around, and finally the squares of each $2 \times 2$ part are moved around. What numbers end up on the main diagonal (from the top left to bottom right)?

$ABCD$ is a parallelogram. $K$ is the circumcircle of $ABD$ . The lines $BC$ and $CD$ meet $K$ again at $E$ and $F$ . Show that the circumcenter of $CEF$ lies on $K$ .

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$ ?

A domino has two numbers (which may be equal) between $0$ and $6$ , one at each end. The domino may be turned around. There is one domino of each type, so $28$ in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino $(3,4)$ , total $3 + 4 = 7$ . Then $(1,3)$ , total $1 + 3 + 3 + 4 = 11$ , then $(4,4)$ , total $11 + 4 + 4 = 19$ . What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

Attachments:

A trio is a set of three distinct integers such that two of the numbers are divisors or multiples of the third. Which trio contained in $\{1, 2, ... , 2002\}$ has the largest possible sum? Find all trios with the maximum sum.

Let $ABCD$ be a quadrilateral with $\measuredangle DAB=\measuredangle ABC=90^{\circ}$ . Denote by $M$ the midpoint of the side $AB$ , and assume that $\measuredangle CMD=90^{\circ}$ . Let $K$ be the foot of the perpendicular from the point $M$ to the line $CD$ . The line $AK$ meets $BD$ at $P$ , and the line $BK$ meets $AC$ at $Q$ . Show that $\angle{AKB}=90^{\circ}$ and $\frac{KP}{PA}+\frac{KQ}{QB}=1$ .

[Moderator edit: The proposed solution can be found at http://erdos.fciencias.unam.mx/mexproblem3.pdf .]