2008 District Round (Round II)

1

Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations: $x^n+sx=2007$ and $x^n+tx=2008$ have at least one common real root.

2

Two circles $U,V$ have distinct radii,tangent to each other externally at $T$.$A,B$ are points on $U,V$ respectively,both distinct from $T$,such that $\angle ATB=90$. (1)Prove that line $AB$ passes through a fixed point; (2)Find the locus of the midpoint of $AB$.

3

For $n>2$, an $n\times n$ grid of squares is coloured black and white like a chessboard, with its upper left corner coloured black. Then we can recolour some of the white squares black in the following way: choose a $2\times 3$ (or $3\times 2$) rectangle which has exactly $3$ white squares and then colour all these $3$ white squares black. Find all $n$ such that after a series of such operations all squares will be black.

4

A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.