1987 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1987 B6

F

be the field of

p^2

elements, where

p

is an odd prime. Suppose

S

(p^2-1)/2

distinct nonzero elements of

F

with the property that for each

a\neq 0

F

, exactly one of

a

-a

S

N

be the number of elements in the intersection

S \cap \{2a: a \in S\}

N

1

Putnam 1987 B5

O_n

n

-dimensional vector

(0,0,\cdots, 0)

M

2n \times n

matrix of complex numbers such that whenever

(z_1, z_2, \dots, z_{2n})M = O_n

z_i

, not all zero, then at least one of the

z_i

is not real. Prove that for arbitrary real numbers

r_1, r_2, \dots, r_{2n}

, there are complex numbers

w_1, w_2, \dots, w_n

\mathrm{re}\left[ M \left( \begin{array}{c} w_1 \\ \vdots \\ w_n \end{array} \right) \right] = \left( \begin{array}{c} r_1 \\ \vdots \\ r_n \end{array} \right).

C

is a matrix of complex numbers,

\mathrm{re}(C)

is the matrix whose entries are the real parts of the entries of

C

1

Putnam 1987 B4

(x_1,y_1) = (0.8, 0.6)

x_{n+1} = x_n \cos y_n - y_n \sin y_n

y_{n+1}= x_n \sin y_n + y_n \cos y_n

n=1,2,3,\dots

\lim_{n\to \infty} x_n

\lim_{n \to \infty} y_n

, prove that the limit exists and find it or prove that the limit does not exist.

1

Putnam 1987 B3

F

be a field in which

1+1 \neq 0

. Show that the set of solutions to the equation

x^2+y^2=1

x

y

F

(x,y)=(1,0)

(x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right)

r

runs through the elements of

F

r^2\neq -1

1

Putnam 1987 B2

r,s

t

be integers with

0 \leq r

0 \leq s

r+s \leq t

\frac{\binom s0}{\binom tr} + \frac{\binom s1}{\binom{t}{r+1}} + \cdots + \frac{\binom ss}{\binom{t}{r+s}} = \frac{t+1}{(t+1-s)\binom{t-s}{r}}.

1

Putnam 1987 B1

\int_2^4 \frac{\sqrt{\ln(9-x)}\,dx}{\sqrt{\ln(9-x)}+\sqrt{\ln(x+3)}}.

1

Putnam 1987 A6

For each positive integer

n

a(n)

be the number of zeroes in the base 3 representation of

n

. For which positive real numbers

x

does the series

\sum_{n=1}^\infty \frac{x^{a(n)}}{n^3}

1

Putnam 1987 A5

\vec{G}(x,y) = \left( \frac{-y}{x^2+4y^2}, \frac{x}{x^2+4y^2},0 \right).

Prove or disprove that there is a vector-valued function

\vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z))

with the following properties:
(i)

M,N,P

have continuous partial derivatives for all

(x,y,z) \neq (0,0,0)

\mathrm{Curl}\,\vec{F} = \vec{0}

(x,y,z) \neq (0,0,0)

\vec{F}(x,y,0) = \vec{G}(x,y)

1

Putnam 1987 A4

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) \mbox{for all real $x,y,z,u$}, and such that

P(1,0,0)=4

P(0,1,0)=5

P(0,0,1)=6

A,B,C

be complex numbers with

P(A,B,C)=0

|B-A|=10

|C-A|

1

Putnam 1987 A3

x

, the real-valued function

y=f(x)

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

f(x)>0

x

f'(x) > 0

x

? Explain. (b) If

f'(x)>0

x

f(x) > 0

x

1

Putnam 1987 A2

The sequence of digits

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots

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)

m

f(2)=2

because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof,

f(1987)

1

Putnam 1987 A1

A,B,C

D

are defined in the plane as follows: \begin{align*} A &= \left\{ (x,y): x^2-y^2 = \frac{x}{x^2+y^2} \right\}, \\ B &= \left\{ (x,y): 2xy + \frac{y}{x^2+y^2} = 3 \right\}, \\ C &= \left\{ (x,y): x^3-3xy^2+3y=1 \right\}, \\ D &= \left\{ (x,y): 3x^2 y - 3x - y^3 = 0\right\}. \end{align*} Prove that

A \cap B = C \cap D