OBM 1985

$a, b, c, d$ are integers with $ad \ne bc$ . Show that $1/((ax+b)(cx+d))$ can be written in the form $r/(ax+b) + s/(cx+d)$ . Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$ .

Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$ .

A convex quadrilateral is inscribed in a circle of radius $1$ . Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$ .

$a, b, c, d$ are integers. Show that $x^2 + ax + b = y^2 + cy + d$ has infinitely many integer solutions iff $a^2 - 4b = c^2 - 4d$ .

$A, B$ are reals. Find a necessary and sufficient condition for $Ax + B[x] = Ay + B[y]$ to have no solutions except $x = y$ .