==> competition/tests/math/putnam/putnam.1992.p <==
Problem A1

Prove that f(n) = 1 - n is the only integer-valued function defined on
the integers that satisfies the following conditions.
(i) f(f(n)) = n, for all integers n;
(ii) f(f(n + 2) + 2) = n for all integers n;
(iii) f(0) = 1.


Problem A2

Define C(alpha) to be the coefficient of x^1992 in the power series
expansion about x = 0 of (1 + x)^alpha. Evaluate

\int_0^1  C(-y-1) ( 1/(y+1) + 1/(y+2) + 1/(y+3) + ... + 1/(y+1992) )  dy.


Problem A3

For a given positive integer m, find all triples (n,x,y) of positive
integers, with n relatively prime to m, which satisfy (x^2 + y^2)^m = (xy)^n.


Problem A4

Let f be an infinitely differentiable real-valued function defined on
the real numbers. If

f(1/n) = n^2/(n^2 + 1), n = 1,2,3,...,

compute the values of the derivatives f^(k)(0), k = 1,2,3,... .


Problem A5

For each positive integer n, let

a_n = {  0 if the number of 1's in the binary representation of n is even,
         1 if the number of 1's in the binary representation of n is odd.

Show that there do not exist positive integers k and m such that

a_{k+j} = a_{k+m+j} = a_{k+2m+j},  for 0 <= j <= m - 1.


Problem A6

Four points are chosen at random on the surface of a sphere. What is the
probability that the center of the sphere lies inside the tetrahedron
whose vertices are at the four points? (It is understood that each point
is independently chosen relative to a uniform distribution on the sphere.)


Problem B1

Let S be a set of n distinct real numbers. Let A_S be the set of numbers
that occur as averages of two distinct elements of S. For a given n >= 2,
what is the smallest possible number of distinct elements in A_S?


Problem B2

For nonnegative integers n and k, define Q(n,k) to be the coefficient of
x^k in the expansion of (1+x+x^2+x^3)^n. Prove that

Q(n,k) = \sum_{j=0}^n {n \choose j} {n \choose k - 2j},

where {a \choose b} is the standard binomial coefficient. (Reminder: For
integers a and b with a >= 0, {a \choose b} = a!/b!(a-b)! for 0 <= b <= a,
and {a \choose b} = 0 otherwise.)


Problem B3

For any pair (x,y) of real numbers, a sequence (a_n(x,y))_{n>=0} is
defined as follows:

a_0(x,y) = x
a_{n+1}(x,y) = ( (a_n(x,y))^2 + y^2 ) / 2, for all n >= 0.

Find the area of the region  { (x,y) | (a_n(x,y))_{n>=0} converges }.


Problem B4

Let p(x) be a nonzero polynomial of degree less than 1992 having no
nonconstant factor in common with x^3 - x. Let

( d^1992 / dx^1992 ) ( p(x) / x^3 - x )  =  f(x) / g(x)

for polynomials f(x) and g(x). Find the smallest possible degree of f(x).


Problem B5

Let D_n denote the value of the (n-1) by (n-1) determinant

| 3 1 1 1 ...  1  |
| 1 4 1 1 ...  1  |
| 1 1 5 1 ...  1  |
| 1 1 1 6 ...  1  |
| . . . . ...  .  |
| 1 1 1 1 ... n+1 |

Is the set {D_n/n!}_{n >= 2} bounded?


Problem B6

Let M be a set of real n by n matrices such that
(i) I \in M, where I is the n by n identity matrix;
(ii) if A \in M and B \in M, then either AB \in M or -AB \in M, but not both;
(iii) if A \in M and B \in M, then either AB = BA or AB = -BA;
(iv) if A \in M and A \noteq I, there is at least one B \in M such that
     AB = -BA.

Prove that M contains at most n^2 matrices.