You should be able to solve this

Round 2, bitches.

These are from a state high school math competition that was held at my uni a while ago.  As far as I know, the problems aren't anywhere on the interbutts.

1)  How many pairs of integers (x,y) satisfy |x|+|y| \le 2008? (Answer should be an integer)

2) Take the integers 1, 2, \ldots ,2008 and write them down in some order to give the sequence a_1, a_2, \ldots, a_2008.  Prove that

|a_1-1| + |a_2-2| + \ldots + |a_2008-2008|

is even.

3) (Some geometry question with a diagram and crap. Forget it.)

4) Find, with proof, all polynomials P(x) such that P(0) = 0 and, for all integers x,

P(x^2+1) = \left( P(x) \right)^2 + 1.

5) A triangle has area 2008 and perimeter 1492.  What is its inradius?

6) P(x) and Q(x) are polynomials with integer coefficients.  There is some integer a such that both a and a+1 are roots of P(x).  In addition you know that Q(2008) = 1776.  Prove that the equation Q\left(P(x)\right) = 1 has no solution.

*crosses fingers*

Maybe I'll do a few later, high school maths doesn't sounds especially appealing.

1)
Well let H_x be the number of y for each x.
H_x = H_-x

so the number we want is 2*Sum H_x

Now H_2008 = 1

Also H_n = H_(n+1) + 2, as any y satisfying n+1, also satisfies n, and then there are two more.

So H is a arithmetic sequence, apply the obvious formula, yada yada.
Most of the shit I've written is pretty obvious as well, the other thread was more fun :(

2) Induction almost certainly does it, might have to show P_n => P_n+2 if it's only true for even numbers in general, can't be arsed to check.


4) Well, P(1) = 1, P(2) = 2, P(5) = 5, P(26) = 26 etc.

But this means that if we take any n, and consider all the polynomials of degree n that satisfy this equation, we can find a sequence of n+1 numbers s.t P(a_n) = a_n for them all.

But two distinct polynomials of degree <= n cannot agree on more than n numbers, so the only possible polynomial is the identity.

5) no idea what inradius means.

6) Hmm, I'll think on it, this one looks more interesting