38th Swedish 1998

1.  Find all positive integers a, b, c, such that (8a-5b)2 + (3b-2c)2 + (3c-7a)2 = 2.
2.  ABC is a triangle. Show that c ≥ (a+b) sin(C/2).
3.  A cube side 5 is made up of unit cubes. Two small cubes are adjacent if they have a common face. Can we start at a cube adjacent to a corner cube and move through all the cubes just once? (The path must always move from a cube to an adjacent cube).
4.  ABCD is a quadrilateral with ∠A = 90o, AD = a, BC = b, AB = h, and area (a+b)h/2. What can we say about ∠B?
5.  Show that for any n > 5 we can find positive integers x1, x2, ... , xn such that 1/x1 + 1/x2 + ... + 1/xn = 1997/1998. Show that in any such equation there must be two of the n numbers with a common divisor (> 1).
6.  Show that for some c > 0, we have |21/3 - m/n| > c/n3 for all integers m, n with n ≥ 1.

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

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© John Scholes
9 Oct 2003
Last corrected/updated 16 Jan 04