Maths Challenge?

[Yuan G]

Here are a few challenging questions that I received for a preparation course before a maths competition called the AIMO. This set was written by MASA. There were 20 questions and I have answered 16 of them. Took ages! I have a few left. See what people make of them. Good luck!

1) Show that 2007/2 - 2006/3 + 2005/4....+ 1/2008 = 1/1005 + 3/1006 + .... 2007/2008.

2) Prove there are no integers x and y such that x^2 + y + 2 and y^2 + 4x are both perfect squares.

3) ABCD is a rectangle with AB=10, BC=8. L is the point on AB with AL=1. M,N,O are points on BC, CD, and DA respectively and LMNO is a rectangle. Find the 2 possible lengths of BM.

These questions are all at year 9/10 level (the competition I'm entering is for year 9/10)

If anyone finds any mistakes in the questions, inform me via e-mail and I will edit it.

Thanks for all positive contributors

Yuan Guo