==> competition/tests/math/putnam/putnam.1987.p <==
WILLIAM LOWELL PUTNAM MATHEMATICAL COMPETITION

FORTY EIGHTH ANNUAL   Saturday, December 5, 1987

Examination A;

Problem A-1
------- ---

Curves A, B, C, and D, are defined in the plane as follows:

A = { (x,y) : x^2 - y^2 = x/(x^2 + y^2) },

B = { (x,y) : 2xy + y/(x^2 + y^2) = 3 },

C = { (x,y) : x^3 - 3xy^2 + 3y = 1 },

D = { (x,y) : 3yx^2 - 3x - y^3 = 0 }.

Prove that the intersection of A and B is equal to the intersection of
C and D.


Problem A-2
------- ---

The sequence of digits

1 2 3 4 5 6 7 8 9 1 0 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 ...

is obtained by writing the positive integers in order.  If the 10^n th
digit in this sequence occurs in the part of the sequence in which the
m-digit numbers are placed, define f(n) to be m.  For example f(2) = 2
because the 100th digit enters the sequence in the placement of the
two-digit integer 55.  Find, with proof, f(1987).


Problem A-3
------- ---

For all real x, the real valued function y = f(x) satisfies

		y'' - 2y' + y = 2e^x.

(a)  If f(x) > 0 for all real x, must f'(x) > 0 for all real x?  Explain.

(b)  If f'(x) > 0 for all real x, must f(x) > 0 for all real x?  Explain.


Problem A-4
------- ---

Let P be a polynomial, with real coefficients, in three variables and F
be a function of two variables such that

	P(ux,uy,uz) = u^2*F(y-x,z-x) for all real x,y,z,u,

and such that P(1,0,0) = 4, P(0,1,0) = 5, and P(0,0,1) = 6.  Also let
A,B,C be complex numbers with P(A,B,C) = 0 and |B-A| = 10.  Find
|C-A|.


Problem A-5
------- ---

    ^
Let G(x,y) = ( -y/(x^2 + 4y^2) , x/(x^2 + 4y^2), 0 ).  Prove or disprove
that there is a vector-valued function

	^
	F(x,y,z) = ( M(x,y,z) , N(x,y,z) , P(x,y,z) )

with the following properties:


	(1)  M,N,P have continuous partial derivatives for all
	       (x,y,z) <> (0,0,0) ;

	          ^   ^
	(2)  Curl F = 0 for all (x,y,z) <> (0,0,0) ;

	     ^          ^
	(3)  F(x,y,0) = G(x,y).


Problem A-6
------- ---

For each positive integer n, let a(n) be the number of zeros in the
base 3 representation of n.  For which positive real numbers x does
the series

			 inf
			-----
			\     x^a(n)
			 >    ------
			/      n^3
			-----
			n = 1

converge?
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Examination B;

Problem B-1
------- ---

	 4/        (ln(9-x))^(1/2)  dx
Evaluate  | --------------------------------- .
	 2/ (ln(9-x))^(1/2) + (ln(x+3))^(1/2)


Problem B-2
------- ---

Let r, s, and t be integers with 0 =< r, 0 =< s, and r+s =< t.

Prove that

	( s )     ( s )     ( s )           ( s )
	( 0 )     ( 1 )     ( 2 )           ( s )          t+1
	----- +  ------- + ------- + ... + ------- = ---------------- .
	( t )    (  t  )   (  t  )         (  t  )   ( t+1-s )( t-s )
	( r )    ( r+1 )   ( r+2 )         ( r+s )            (  r  )


        ( n )                                  n(n-1)...(n+1-k)
( Note: ( k ) denotes the binomial coefficient ---------------- .)
                                                k(k-1)...3*2*1


Problem B-3
------- ---

Let F be a field in which 1+1 <> 0.  Show that the set of solutions to
the equation x^2 + y^2 = 1 with x and y in F is given by

			    (x,y) = (1,0)

			              r^2 - 1     2r
			and (x,y) = ( ------- , ------- ),
			              r^2 + 1   r^2 + 1

where r runs through the elements of F such that r^2 <> -1.


Problem B-4
------- ---

Let ( x(1), y(1) ) = (0.8,0.6) and let

		    x(n+1) = x(n)*cos(y(n)) - y(n)*sin(y(n))

		and y(n+1) = x(n)*sin(y(n)) + y(n)*cos(y(n))

for n = 1,2,3,...  .  For each of the limits as n --> infinity of
x(n) and y(n), prove that the limit exists and find it or prove that
the limit does not exist.


Problem B-5
------- ---

Let O(n) be the n-dimensional zero vector (0,0,...,0).  Let M be a
2n x n matrix of complex numbers such that whenever
( z(1), z(2), ..., z(2n)*M = O(n), with complex z(i), not all zero,
then at least one of the z(i) is not real.  Prove that for arbitrary
real number r(1), r(2), ..., r(2n), there are complex numbers w(1),
w(2), ..., w(n) such that

		   (   ( w(1) ) )   ( r(1)  )
		   (   (  .   ) )   (   .   )
		Re ( M*(  .   ) ) = (   .   )  .
		   (   (  .   ) )   (   .   )
		   (   ( w(n) ) )   ( r(2n) )

(Note:  If C is a matrix of complex numbers, Re(C) is the matrix whose
entries are the real parts of entries of C.)


Problem B-6
------- ---

Let F be the field of p^2 elements where p is an odd prime.  Suppose S
is a set of (p^2-1)/2 distinct nonzero elements of F with the property
that for each a <> 0 in F, exactly one of a and -a is in S.  Let N be
the number of elements in the intersection of S with { 2a : a e S }.
Prove that N is even.
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