37th Swedish 1997

1.  AC is a diameter of a circle. AB is a tangent. BC meets the circle again at D. AC = 1, AB = a, CD = b. Show that 1/(a2 + ½) < b/a < 1/a2.
2.  ABC is a triangle. BD is the angle bisector. The point E on AB is such that ∠ACE = (2/5) ∠ ACB. BD and CE meet at P. ED = DC = CP. Find the angles of ABC.
3.  a, b are integers with odd sum. Show that every integer can be written as x2 - y2 + ax + by for some integers x, y.
4.  A and B play a game as follows. Each throws a dice. Suppose A gets x and B gets y. If x and y have the same parity, then A wins. If not, then they make a list of all two digit numbers ab ≤ xy with 1 ≤ a, b ≤ 6. Then they take turns (starting with A) replacing two numbers on the list by their (non-negative) difference. When just one number remains, it is compared to x. If it has the same parity A wins, otherwise B wins. Find the probability that A wins.
5.  Let s(n) be the sum of the digits of n. Show that for n > 1 and ≠ 10, there is a unique integer f(n) ≥ 2 such that s(k) + s(f(n) - k) = n for each 0 < k < f(n).
6.  M is a union of finitely many disjoint intervals with total length > 1. Show that there are two distinct points in M whose difference is an integer.

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

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© John Scholes
9 October 2003
Last corrected/updated 19 Jan 04