33rd CanMO 2001


1. A quadratic with integral coefficients has two distinct positive integers as roots, the sum of its coefficients is prime and it takes the value -55 for some integer. Show that one root is 2 and find the other root.
2. The numbers -10, -9, -8, ... , 9, 10 are arranged in a line. A player places a token on the 0 and throws a fair coin 10 times. For each head the token is moved one place to the left and for each tail it is moved one place to the left. If we color one or more numbers black and the remainder white, we find that the chance of the token ending up on a black number is m/n with m + n = 2001. What is the largest possible total for the black numbers?
3. The triangle ABC has AB and AC unequal. The angle bisector of A meets the perpendicular bisector of BC at X. The line joining the feet of the perpendiculars from X to AB and AC meets BC at D. Find BD/DC.
4. A rectangular table has every entry a positive integer. n is a fixed positive integer. A move consists of either subtracting n from every element in a column or multiplying every element in a row by n. Find all n such that we can always end up with all zeros whatever the size or content of the starting table.
5. A₀, A₁, A₂ lie on a circle radius 1 and A₁A₂ is not a diameter. The sequence A_n is defined by the statement that A_n is the circumcenter of A_n-1A_n-2A_n-3. Show that A₁, A₅, A₉, A₁₃, ... are collinear. Find all A₁A₂ for which A₁A₁₀₀₁/A₁₀₀₁A₂₀₀₁ is the 500th power of an integer.

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