OMM 1993

$ABC$ is a triangle with $\angle A = 90^o$ . Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$ . Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$ . $O$ is the midpoint of $BC$ . Let the lines $OD$ and $EC$ meet at $D'$ , and the lines $OE$ and $BD$ meet at $E'$ . Find area $DED'E'$ in terms of the sides of $ABC$ .

Find all numbers between $100$ and $999$ which equal the sum of the cubes of their digits.

Given a pentagon of area $1993$ and $995$ points inside the pentagon, let $S$ be the set containing the vertices of the pentagon and the $995$ points. Show that we can find three points of $S$ which form a triangle of area $\le 1$ .

$f(n,k)$ is defined by (1) $f(n,0) = f(n,n) = 1$ and (2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$ . How many times do we need to use (2) to find $f(3991,1993)$ ?

$OA, OB, OC$ are three chords of a circle. The circles with diameters $OA, OB$ meet again at $Z$ , the circles with diameters $OB, OC$ meet again at $X$ , and the circles with diameters $OC, OA$ meet again at $Y$ . Show that $X, Y, Z$ are collinear.

$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$ .