

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 a^{2} + b^{2} + x^{2} = y^{2} has an integral solution x, y.


4. * is a realvalued operation on the nonzero 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 sa, sb, sc, 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 x_{1}, x_{2}, x_{3} such that f(x_{1}) = x_{2}, f(x_{2}) = x_{3}, f(x_{3}) = x_{1}.

