24th Swedish 1984

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1.  A, B are points inside a circle C. Show that there is a circle through A and B which lies entirely inside C.
2.  Each point in a 3 x 7 array is colored yellow or blue. Show that it is always possible to find 4 points of the same color forming a rectangle with sides parallel to the side of the array.
3.  Show that ( (a+1)/(b+1) )b+1 ≥ (a/b)b for positive reals a, b.
4.  Find all positive integers m, n such that all roots of (x2 - mx + n)(x2 - nx + m) are positive integers.
5.  Find all positive integer solutions to a3 - b3 - c3 = 3abc, a2 = 2(a + b + c).
6.  The positive integers a1, a2, ... , a14 satisfy ∑ 3ai = 6558. Show that they must be two copies of each of 1, 2, ... , 7.

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
8 October 2003
Last corrected/updated 3 Mar 04