

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/(a^{2} + ½) < b/a < 1/a^{2}.


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 x^{2}  y^{2} + 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 (nonnegative) 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.

