AT-OEMO-REG 2007

Let $0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $(x_i,x_j)$ with $0<x_ix_j(x_j-x_i)<\frac{1}{2007}$

Find all tuples $(x_1,x_2,x_3,x_4,x_5)$ of positive integers with $x_1>x_2>x_3>x_4>x_5>0$ and ${\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.$

Let $a$ be a positive real number and $n$ a non-negative integer. Determine $S-T$ , where $S= \sum_{k=-2n}^{2n+1} \frac{(k-1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$ and $T= \sum_{k=-2n}^{2n+1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}$

Let $M$ be the intersection of the diagonals of a convex quadrilateral $ABCD$ . Determine all such quadrilaterals for which there exists a line $g$ that passes through $M$ and intersects the side $AB$ in $P$ and the side $CD$ in $Q$ , such that the four triangles $APM$ , $BPM$ , $CQM$ , $DQM$ are similar.