

1. The real numbers w, x, y, z are all nonnegative and satisfy: y = x  2003, z = 2y  2003, w = 3z  2003. Find the solution with the smallest x.


2. A lecture room has a rectangular array of chairs. There are 6 boys in each row and 8 girls in each column. 15 chairs are unoccupied. What can be said about the number of rows and columns?


3. Which reals x satisfy [x^{2}  2x] + 2[x] = [x]^{2}?


4. Find all real polynomials p(x) such that 1 + p(x) ≡ (p(x1) + p(x+1) )/2.


5. Given two positive reals a, b, how many nonsimilar plane quadrilaterals ABCD have AB = a, BC = CD = DA = b and ∠B = 90^{o}?


6. A sheet of squared paper is infinite in all directions. Each square contains an integer which is the sum of the integers in the squares above and to the left. A particular row, R, has every integer positive. Let R_{n} be the row n below R. Show that there are at most n zeros in R_{n}.

