| 1. Find all non-empty finite sets S of positive integers such that if m,n ∈ S, then (m+n)/gcd(m,n) ∈ S. |
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| 2. ABC is an acute-angled triangle with circumcenter O and orthocenter H (and O ≠ H). Show that one of area AOH, area BOH, area COH is the sum of the other two. |
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| 3. 2004 points are in the plane, no three collinear. S is the set of lines through any two of the points. Show that the points can be colored with two colors so that any two of the points have the same color iff there are an odd number of lines in S which separate them (a line separates them if they are on opposite sides of it). |
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| 4. Show that [(n-1)!/(n2+n)] is even for any positive integer n. |
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| 5. Show that (x2 + 2)(y2 + 2)(z2 + 2) ≥ 9(xy + yz + zx) for any positive reals x, y, z. |
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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
22 March 2004
Last corrected/updated 22 Mar 04