7th Brasil 1985


1. a, b, c, d are integers with ad ≠ bc. Show that 1/((ax+b)(cx+d)) can be written in the form r/(ax+b) + s/(cx+d). Find the sum 1/1·4 + 1/4·7 + 1/7·10 + ... + 1/2998·3001.
2. Given n points in the plane, show that we can always find three which give an angle ≤ π/n.
3. 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.
4. a, b, c, d are integers. Show that x² + ax + b = y² + cy + d has infinitely many integer solutions iff a² - 4b = c² - 4d.
5. A, B are reals. Find a necessary and sufficient condition for Ax + B[x] = Ay + B[y] to have no solutions except x = y.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.