==> competition/tests/math/putnam/putnam.1988.p <==
Problem A-1: Let R be the region consisting of the points (x,y) of the
cartesian plane satisfying both |x| - |y| <= 1 and |y| <= 1.  Sketch
the region R and find its area.

Problem A-2: A not uncommon calculus mistake is to believe that the
product rule for derivatives says that (fg)' = f'g'.  If f(x) =
exp(x^2), determine, with proof, whether there exists an open interval
(a,b) and a non-zero function g defined on (a,b) such that the wrong
product rule is true for x in (a,b).

Problem A-3: Determine, with proof, the set of real numbers x for
which  sum from n=1 to infinity of ((1/n)csc(1/n) - 1)^x  converges.

Problem A-4:
(a) If every point on the plane is painted one of three colors, do
there necessarily exist two points of the same color exactly one inch
apart?
(b) What if "three" is replaced by "nine"?
Justify your answers.

Problem A-5: Prove that there exists a *unique* function f from the
set R+ of positive real numbers to R+ such that
	f(f(x)) = 6x - f(x)  and  f(x) > 0  for all x > 0.

Problem A-6: If a linear transformation A on an n-dimensional vector
space has n + 1 eigenvectors such that any n of them are linearly
independent, does it follow that A is a scalar multiple of the
identity?  Prove your answer.

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Problem B-1: A *composite* (positive integer) is a product ab with a
and b not necessarily distinct integers in {2,3,4,...}.  Show that
every composite is expressible as xy + xz + yz + 1, with x, y, and z
positive integers.

Problem B-2: Prove or disprove: If x and y are real numbers with y >= 0
and y(y+1) <= (x+1)^2, then y(y-1) <= x^2.

Problem B-3: For every n in the set Z+ = {1,2,...} of positive
integers, let r(n) be the minimum value of |c-d sqrt(3)| for all
nonnegative integers c and d with c + d = n.  Find, with proof, the
smallest positive real number g with r(n) <= g for all n in Z+.

Problem B-4: Prove that if  sum from n=1 to infinity a(n)  is a
convergent series of positive real numbers, then so is
sum from n=1 to infinity (a(n))^(n/(n+1)).

Problem B-5: For positive integers n, let M(n) be the 2n + 1 by 2n + 1
skew-symmetric matrix for which each entry in the first n subdiagonals
below the main diagonal is 1 and each of the remaining entries below
the main diagonal is -1.  Find, with proof, the rank of M(n).
(According to the definition the rank of a matrix is the largest k
such that there is a k x k submatrix with non-zero determinant.)

One may note that M(1) = (  0 -1  1 ) and M(2) = (  0 -1 -1  1  1 )
                         (  1  0 -1 )            (  1  0 -1 -1  1 )
                         ( -1  1  0 )            (  1  1  0 -1 -1 )
                                                 ( -1  1  1  0 -1 )
                                                 ( -1 -1  1  1  0 )

Problem B-6: Prove that there exist an infinite number of ordered
pairs (a,b) of integers such that for every positive integer t the
number at + b is a triangular number if and only if t is a triangular
number.  (The triangular numbers are the t(n) = n(n + 1)/2 with n in
{0,1,2,...}.)