OMM 1996

Let $P$ and $Q$ be the points on the diagonal $BD$ of a quadrilateral $ABCD$ such that $BP = PQ = QD$ . Let $AP$ and $BC$ meet at $E$ , and let $AQ$ meet $DC$ at $F$ . (a) Prove that if $ABCD$ is a parallelogram, then $E$ and $F$ are the midpoints of the corresponding sides. (b) Prove the converse of (a).

There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move simultaneously in a circular way (following the numbering sense) as follows: chip $1$ moves one booth, chip $2$ moves two booths, etc., so that more than one chip can be in the same booth. At any minute, for each chip sharing a booth with chip $1$ a bulb is lit. Where is chip $1$ on the first minute in which all bulbs are lit?

Prove that it is not possible to cover a $6\times 6$ square board with eighteen $2\times 1$ rectangles, in such a way that each of the lines going along the interior gridlines cuts at least one of the rectangles. Show also that it is possible to cover a $6\times 5$ rectangle with fifteen $2\times 1$ rectangles so that the above condition is fulfilled.

For which integers $n\ge 2$ can the numbers $1$ to $16$ be written each in one square of a squared $4\times 4$ paper such that the $8$ sums of the numbers in rows and columns are all different and divisible by $n$ ?

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$ , let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$ .

In a triangle $ABC$ with $AB < BC < AC$ , points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp CA$ and $BB'=CA$ , and $CC' \perp AB$ and $CC'= AB$ , as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C'$ are collinear.

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