OMM 1994

The sequence $1, 2, 4, 5, 7, 9 ,10, 12, 14, 16, 17, ...$ is formed as follows. First we take one odd number, then two even numbers, then three odd numbers, then four even numbers, and so on. Find the number in the sequence which is closest to $1994$ .

The $12$ numbers on a clock face are rearranged. Show that we can still find three adjacent numbers whose sum is $21$ or more.

$ABCD$ is a parallelogram. Take $E$ on the line $AB$ so that $BE = BC$ and $B$ lies between $A$ and $E$ . Let the line through $C$ perpendicular to $BD$ and the line through $E$ perpendicular to $AB$ meet at $F$ . Show that $\angle DAF = \angle BAF$ .

A capricious mathematician writes a book with pages numbered from $2$ to $400$ . The pages are to be read in the following order. Take the last unread page ( $400$ ), then read (in the usual order) all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. So, the order would be $2, 4, 5, ... , 400, 3, 7, 9, ... , 399, ...$ . What is the last page to be read?

$ABCD$ is a convex quadrilateral. Take the $12$ points which are the feet of the altitudes in the triangles $ABC, BCD, CDA, DAB$ . Show that at least one of these points must lie on the sides of $ABCD$ .

Show that we cannot tile a $10 x 10$ board with $25$ pieces of type $A$ , or with $25$ pieces of type $B$ , or with $25$ pieces of type $C$ .

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