### 33rd Swedish 1993

 1.  n and 3n have the same digit sum. Show that n is a multiple of 9. 2.  A railway line passes through (in order) the 11 stations A, B, ... , K. The distance from A to K is 56. The distances AC, BD, CE, ... , IK are each ≤ 12. The distances AD, BE, CF, ... , HK are each ≥ 17. Find the distance between B and G. 3.  Show that if ab are integers with ab even, then a2 + b2 + x2 = y2 has an integral solution x, y. 4.  * is a real-valued operation on the non-zero reals which satisfies (1) a * a = 1, (2) a * (b * c) = (a * b) c (in other words, the ordinary product of (a * b) and c), for all a, b, c. Solve x * 36 = 216. 5.  Given a triangle with sides a, b, c, we form, if we can, a triangle with sides s-a, s-b, s-c, where s = (a+b+c)/2. For which triangles can this be repeated indefinitely? 6.  For reals a, b define the function f(x) = 1/(ax+b). For which a, b are there distinct reals x1, x2, x3 such that f(x1) = x2, f(x2) = x3, f(x3) = x1.

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

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