SE-SKOLORNAS 2000

Each of the numbers $1, 2, ... , 10$ is colored red or blue. $5$ is red and at least one number is blue. If $m, n$ are different colors and $m+n \le 10$ , then $m+n$ is blue. If $m, n$ are different colors and $mn \le 10$ , then $mn$ is red. Find all the colors.

$p(x)$ is a polynomial such that $p(y^2+1) = 6y^4 - y^2 + 5$ . Find $p(y^2-1)$ .

Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

The vertices of a triangle are three-dimensional lattice points. Show that its area is at least $\frac12$ .

Let $f(n)$ be defined on the positive integers and satisfy: $f(prime) = 1$ , $f(ab) = a f(b) + f(a) b$ . Show that $f$ is unique and find all $n$ such that $n = f(n)$ .

Solve $$\left\{ \begin{array}{l} y(x+y)^2 = 9 y(x^3-y^3) = 7 \end{array} \right.$$