43rd Swedish 2003

1.  The real numbers w, x, y, z are all non-negative 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 [x2 - 2x] + 2[x] = [x]2?
4.  Find all real polynomials p(x) such that 1 + p(x) ≡ (p(x-1) + p(x+1) )/2.
5.  Given two positive reals a, b, how many non-similar plane quadrilaterals ABCD have AB = a, BC = CD = DA = b and ∠B = 90o?
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 Rn be the row n below R. Show that there are at most n zeros in Rn.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.

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© John Scholes
11 January 2004
Last corrected/updated 13 Jan 04