Here are the questions to the 62nd annual Putnam exam, held today 
(Dec 1 2001).  You have 6 hours; good luck :-)

I will post the answers I have as soon as I can type them up. 
(Right now I lack answers to  A6, B5, B6 .)

There are links to Putnam problems and solutions at
	http://www.math.niu.edu/~rusin/problems-math/



A1. Consider a set  S  and a binary operation  *  on  S  (that is, for
each  a, b  in  S,  a*b  is in  S).  Assume that  (a*b)*a = b  for all
a, b  in  S.  Prove that  a*(b*a) =b  for all a, b  in  S.

A2. You have coins  C1, C2, ..., C_n.  For each  k,  coin  C_k  is biased
so that, when tossed, it has probability  1/(2k+1)  of falling heads.
If the  n  coins are tossed, what is the probability that the
number of heads is odd? Express the answer as a rational function of  n.

A3. For each integer  m,  consider the polynomial
	P_m(x) = x^4 - (2m+4) x^2 + (m-2)^2
For what values of  m  is  P_m(x)  the product of two nonconstant 
polynomials with integer coefficients?

A4. Triangle  ABC  has area  1.  Points  E,F,G  lie, respectively, on
sides  BC, CA, AB  such that  AE  bisects  BF  at point  R, 
BF bisects  CG  at point  S,  and  CG  bisects  AE  at point  T.
Find the area of triangle  RST.
[Illustration deleted.]

A5. Prove that there are unique positive integers  a, n  such that
	a^(n+1) - (a+1)^n = 2001.

A6. Can an arc of a parabola inside a circle of radius  1 have length
greater than  4 ?

B1. Let  n  be an even positive integer. Write the numbers  1, 2, ..., n^2
in the squares of an  n x n  grid so that the  k-th row, from left to
right, is  
	(k-1)n + 1,  (k-1)n + 2, ..., (k-1)n + n.
Color the squares of the grid so that half of the squares in each row
and in each column are red and the other half are black (a checkerboard
coloring is one possibility). Prove that for each such coloring, the
sum of the numbers on the red squares is equal to the sum of the numbers
on the black squares.

B2. Find all pairs of real numbers  (x,y)  satisfying the system of equations

	1/x + 1/(2y) = (x^2 + 3 y^2) ( 3 x^2 + y^2 )
	1/x - 1/(2y) = 2(y^4 - x^4)

B3. For any positive integer  n  let  <n>  denote the closest integer
to  sqrt(n).  Evaluate
	\sum_{n=1}^{\infty}  ( 2^{<n>} + 2^{-<n>} ) / 2^n

B4. Let  S  denote the set of rational numbers different from -1, 0, and 1.
Define  f : S --> S  by  f(x) = x - 1/x . Prove or disprove that
	\intersect_{n=1}^{\infty}  f^(n) (S) = \emptyset,
where  f^(n) = f o f o  ... o f   (n  times).

(Note:  f(S)  denotes the set of all values  f(s)  for  s \in  S. )

B5. Let  a  and  b  be real numbers in the interval  (0, 1/2)  and
let  g  be a continuous real-valued function such that 
g(g(x)) = a g(x) + b x  for all real  x.  Prove that  g(x) = c x  for
some constant  c.

B6. Assume that  (a_n)_{n >= 1}  is an increasing sequence of positive
real numbers such that  lim_{n->\infty}  a_n / n = 0.  Must there
exist infinitely many positive integers  n  such that
	a_{n-i} + a_{n+i} < 2 a_n  for  i = 1, 2, ..., n-1 ?