OMM 1992

The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$ . $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.

Given a prime number $p$ , how many $4$ -tuples $(a, b, c, d)$ of positive integers with $0 \le a, b, c, d \le p-1$ satisfy $ad = bc$ mod $p$ ?

Given $7$ points inside or on a regular hexagon, show that three of them form a triangle with area $\le 1/6$ the area of the hexagon.

Show that $1 + 11^{11} + 111^{111} + 1111^{1111} +...+ 1111111111^{1111111111}$ is divisible by $100$ .

$x, y, z$ are positive reals with sum $3$ . Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$

$ABCD$ is a rectangle. $I$ is the midpoint of $CD$ . $BI$ meets $AC$ at $M$ . Show that the line $DM$ passes through the midpoint of $BC$ . $E$ is a point outside the rectangle such that $AE = BE$ and $\angle AEB = 90^o$ . If $BE = BC = x$ , show that $EM$ bisects $\angle AMB$ . Find the area of $AEBM$ in terms of $x$ .