

1. 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 ≤ 10, then m+n is blue. If m, n are different colors and mn ≤ 10, then mn is red. Find all the colors.


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


3. Are there any integral solutions to n^{2} + (n+1)^{2} + (n+2)^{2} = m^{2}?


4. The vertices of a triangle are threedimensional lattice points. Show that its area is at least ½.


5. 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).


6. Solve y(x+y)^{2} = 9, y(x^{3}y^{3}) = 7.

