==> competition/tests/math/putnam/putnam.1990.p <==
Problem A-1
How many primes among the positive integers, written as usual in base
10, are such that their digits are alternating 1's and 0's, beginning
and ending with 1?

Problem A-2
Evaluate \int_0^a \int_0^b \exp(\max{b^2 x^2, a^2 y^2}) dy dx where a and
b are positive.

Problem A-3
Prove that if 11z^10 + 10iz^9 + 10iz - 11 = 0, then |z| = 1. (Here z is
a complex number and i^2 = -1.)

Problem A-4
If \alpha is an irrational number, 0 < \alpha < 1, is there a finite
game with an honest coin such that the probability of one player winning
the game is \alpha? (An honest coin is one for which the probability of
heads and the probability of tails are both 1/2. A game is finite if
with probability 1 it must end in a finite number of moves.)

Problem A-5
Let m be a positive integer and let G be a regular (2m + 1)-gon
inscribed in the unit circle. Show that there is a positive constant A,
independent of m, with the following property. For any point p inside G
there are two distinct vertices v_1 and v_2 of G such that
                             1     A
| |p - v_1| - |p - v_2| | < --- - ---.
                             m    m^3
Here |s - t| denotes the distance between the points s and t.

Problem A-6
Let \alpha = 1 + a_1 x + a_2 x^2 + ... be a formal power series with
coefficients in the field of two elements. Let
          / 1 if every block of zeros in the binary expansion of n
	 /    has an even number of zeros in the block,
  a_n = {
	 \ 0 otherwise.
(For example, a_36 = 1 because 36 = 100100_2 and a_20 = 0 because 20 =
10100_2.) Prove that \alpha^3 + x \alpha + 1 = 0.

Problem B-1
A dart, thrown at random, hits a square target. Assuming that any two
points of the target of equal area are equally likely to be hit, find
the probability that the point hit is nearer to the center than to any
edge. Express your answer in the form (a \sqrt(b) + c) / d, where a, b,
c, d are integers.

Problem B-2
Let S be a non-empty set with an associative operation that is left and
right cancellative (xy = xz implies y = z, and yx = zx implies y = z).
Assume that for every a in S the set { a^n : n = 1, 2, 3, ... } is
finite. Must S be a group?

Problem B-3
Let f be a function on [0,\infty), differentiable and satisfying
  f'(x) = -3 f(x) + 6 f(2x)
for x > 0. Assume that |f(x)| <= \exp(-\sqrt(x)) for x >= 0 (so that
f(x) tends rapidly to 0 as x increases). For n a non-negative integer,
define
  \mu_n = \int_0^\infty x^n f(x) dx
(sometimes called the nth moment of f).
a. Express \mu_n in terms of \mu_0.
b. Prove that the sequence { \mu_n 3^n / n! } always converges, and that
   the limit is 0 only if \mu_0 = 0.

Problem B-4
Can a countably infinite set have an uncountable collection of non-empty
subsets such that the intersection of any two of them is finite?

Problem B-5
Label the vertices of a trapezoid T (quadrilateral with two parallel
sides) inscribed in the unit circle as A, B, C, D so that AB is parallel
to CD and A, B, C, D are in counterclockwise order. Let s_1, s_2, and d
denote the lengths of the line segments AB, CD, and OE, where E is the
point of intersection of the diagonals of T, and O is the center of the
circle. Determine the least upper bound of (s_1 - s_2) / d over all such
T for which d \ne 0, and describe all cases, if any, in which it is
attained.

Problem B-6
Let (x_1, x_2, ..., x_n) be a point chosen at random from the
n-dimensional region defined by 0 < x_1 < x_2 < ... < x_n < 1.
Let f be a continuous function on [0, 1] with f(1) = 0. Set x_0 = 0 and
x_{n+1} = 1. Show that the expected value of the Riemann sum
  \sum_{i = 0}^n (x_{i+1} - x_i) f(x_{i+1})
is \int_0^1 f(t)P(t) dt, where P is a polynomial of degree n,
independent of f, with 0 \le P(t) \le 1 for 0 \le t \le 1.