A1. Given a set X with a binary operation *, not necessarily associative or
commutative,but such that (x * y) * x = y for all x, y in X.
Show that x * (y * x) = y for all x, y in X.
A2. You have a set of n biased coins. The mth coin has probability
1/(2m+1) of landing heads (m = 1, 2, ... , n)
and the results for each coin are independent.
What is the probability that if each coin is tossed once,
you get an odd number of heads?
A3. For what integers n is the polynomial x4 - (2n + 4) x2 + (n - 2)2
the product of two non-trivial polynomials with integer coefficients?
A4. Points X, Y, Z lie on the sides BC, CA, AB (respectively) of
the triangle ABC. AX bisects BY at K, BY bisects CZ at L, CZ bisects AX at M.
Find the area of the triangle KLM as a fraction of that of ABC.
A5. Find all solutions to xn+1 - (x + 1)n = 2001 in
positive integers x,n.
A6. A parabola intersects a disk radius 1. Can the arc length of
the parabol a inside the disk exceed 4? |
