OBM 1984

Find all solutions in positive integers to $(n+1)^k -1 = n!$

Each day $289$ students are divided into $17$ groups of $17$ . No two students are ever in the same group more than once. What is the largest number of days that this can be done?

Given a regular dodecahedron of side $a$ . Take two pairs of opposite faces: $E, E'$ and $F, F'$ . For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.

$ABC$ is a triangle with $\angle A = 90^o$ . For a point $D$ on the side $BC$ , the feet of the perpendiculars to $AB$ and $AC$ are $E$ and $F$ . For which point $D$ is $EF$ a minimum?

$ABCD$ is any convex quadrilateral. Squares center $E, F, G, H$ are constructed on the outside of the edges $AB, BC, CD$ and $DA$ respectively. Show that $EG$ and $FH$ are equal and perpendicular.

There is a piece on each square of the solitaire board shown except for the central square. A move can be made when there are three adjacent squares in a horizontal or vertical line with two adjacent squares occupied and the third square vacant. The move is to remove the two pieces from the occupied squares and to place a piece on the third square. (One can regard one of the pieces as hopping over the other and taking it.) Is it possible to end up with a single piece on the board, on the square marked $X$ ?

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