Putnam exam, Dec 2 2000.
Better presentation, and a discussion of some solutions, at
	http://www.math.niu.edu/~rusin/problems-math/


Problem A1

Let  A  be a positive real number. What are the possible values of
Sum_{j=0}^\infty x_j^2 ,  given that  x_0, x_1, x_2,... 
are positive numbers for which   \sum_{j=0}^\infty x_j = A ?


Problem A2

Prove that there exist infinitely many integers  n  such that
n ,  n+1 , and  n+2  are each the sum of two squares of integers.

[Example:  0=0^2+0^2 ,  1=0^2+1^2 , and  2=1^2+1^2 .]


Problem A3

The octagon  P_1P_2P_3P_4P_5P_6P_7P_8  is inscribed in a circle, with the
vertices around the circumference in the given order. Given that the polygon
P_1P_3P_5P_7  is a square of area 5 and the polygon  P_2P_4P_6P_8  is a
rectangle of area 4, find the maximum possible area of the octagon.


Problem A4

Show that the improper integral
          \lim_{B -> \infty} \int_0^B sin(x) sin(x^2) dx  
converges.


Problem A5

Three distinct points with integer coordinates lie in the plane on a circle of
radius   r>0 .  Show that two of these points are separated by a distance of
at least  r^{1/3} .


Problem A6

Let  f(x)  be a polynomial with integer coefficients. Define a sequence
a_0, a_1, ...  of integers such that  a_0=0  and  a_{n+1}=f(a_n)  for
all  n >=  0 . Prove that if there exists a positive integer  m  for which
a_m=0  then either  a_1=0  or  a_2=0 .


Problem B1

Let  a_j ,  b_j , and  c_j  be integers for  1 <=  j  <=  N . Assume, for each
 j , that at least one of  a_j ,  b_j ,  c_j  is odd. Show that there exist
integers  r ,  s ,  t  such that  r a_j + s b_j + t c_j  is odd for at least
4N/7  values of  j ,  1 <=  j  <=  N .


Problem B2

Prove that the expression
               ( gcd(m,n) / n ) binomial(n,m)   
is an integer for all pairs of integers  n >=  m >=  1 . [Here  binomial(n,m)=
n!/(m!(n-m)!)  and   gcd(m,n)  is the greatest common divisor of  m  and  n .]


Problem B3

Let   f(t) = \sum_{j=1}^N a_j sin(2\pi j t) , where each  a_j  is real and  
a_N <> 0 . Let  N_k  denote the number of zeros (including multiplicities) of
(d^k f)/(dt^k) . Prove that
   N_0 <=  N_1 <=  N_2 <=  ...     and  \lim_{k -> \infty} N_k = 2N.  


Problem B4

Let  f(x)  be a continuous function such that  f( 2x^2-1 ) = 2 x f(x)  for 
all  x . Show that  f(x)=0  for  -1  <=  x  <=  1 .


Problem B5

Let  S_0  be a finite set of positive integers. We define finite sets 
S_1, S_2, ...  of positive integers as follows:

        Integer  a  is in  S_{n+1}  if and only if 
                     exactly one of  a-1  or  a  is in  S_n .

Show that there exist infinitely many integers  N  for which
S_N = S_0  union  { N + a : a  in  S_0 } .


Problem B6

Let  B  be a set of more than  2^{n+1}/n  distinct points with
coordinates of the form  (+- 1, +- 1, ..., +- 1)  in  n-dimensional 
space, with  n  >=  3 . Show that there are three distinct
points in  B  which are the vertices of an equilateral triangle.