| A1. Find the largest k such that for every positive integer n we can find at least k numbers in the set {n+1, n+2, ... , n+16} which are coprime with n(n+17). | |
| A2. Given a square side 1 and 2n positive reals a1, b1, ... , an, bn each ≤ 1 and satisfying ∑ aibi ≥ 100. Show that the square can be covered with rectangles Ri with sides length (ai, bi) parallel to the square sides. | |
| A3. The function f : R → R satisfies f(3x) = 3f(x) - 4f(x)3 for all real x and is continuous at x = 0. Show that |f(x)| ≤ 1 for all x. | |
| B1. P is a point inside the triangle ABC is a triangle. The distance of P from the lines BC, CA, AB is da, db, dc respectively. Show that 2/(1/da + 1/db + 1/dc) < r < (da + db + dc)/2, where r is the inradius. | |
| B2. p(x,y) is a polynomial such that p(cos t, sin t) = 0 for all real t. Show that there is a polynomial q(x,y) such that p(x,y) = (x2 + y2 - 1) q(x,y). | |
| B3. There is a convex polyhedron with k faces. Show that if k/2 of the faces are such that no two have a common edge, then the polyhedron cannot have an inscribed sphere. |
To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.
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© John Scholes
jscholes@kalva.demon.co.uk
20 March 2004
Last corrected/updated 24 Mar 04