62nd Putnam 2001

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 parabola inside the disk exceed 4?
B1.  Number the cells in a 2n x 2n grid from 1 to 4n2, starting at the top left and moving left to right along each row, then continuing at the left of the next row down and so on. Colour half the cells black and half white, so that just half the cells in each row and half the cells in each column are black. Show that however this is done the sum of the black squares equals the sum of the white squares.
B2.  Find all real solutions (x, y) to the simultaneous equations:
    1/x - 1/(2y) = 2 y4 - 2 x4
    1/x + 1/(2y) = (3 x2 + y2)(x2 + 3 y2).
B3.  Let @n denote the closest integer to √n. Find ∑ (2@n + 2-@n)/2n, where the sum is taken from n = 1 to n = infinity.
B4.  Let X be the set of rationals excluding 0, ±1. Let f: X → X be defined as f(x) = x - 1/x. Let f1(x) = f(x), f2(x) = f(f(x)), f3(x) = f(f2(x)) etc. Does there exist a value x in X such that for any positive integer n, we can find y in X with fn(y) = x?
B5.  f is a continuous real-valued function on the reals such that for some 0 < a, b < 1/2, we have f(f(x)) = a f(x) + b x for all x. Show that for some constant k, f(x) = k x for all x.
B6.  x1 < x2 < x3 < ... is a sequence of positive reals such that lim xn/n = 0. Is it true that we can find arbitrarily large N such that all of (x1 + x2N-1), (x2 + x2N-2), (x3 + x2N-3), ... , (xN-1 + xN+1) are less than 2 xN?

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

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© John Scholes
16 Dec 2001