1947 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1947 B6

OX, OY, OZ

be mutually orthogonal lines in space. Let

C

be a fixed point on

OZ

U,V

variable points on

OX, OY,

respectively. Find the locus of a point

P

PU, PV, PC

are mutually orthogonal.

1

Putnam 1947 B5

a,b,c,d

be distinct integers such that

(x-a)(x-b)(x-c)(x-d) -4=0

has an integer root

r.

4r=a+b+c+d.

1

Putnam 1947 B4

P(z)= z^2 +az +b,

a,b \in \mathbb{C}.

|P(z)|=1

for every complex number

z

|z|=1.

a=b=0.

1

Putnam 1947 B3

x,y

be cartesian coordinates in the plane.

I

denotes the line segment

1\leq x\leq 3 , y=1.

For every point

P

I

P'

denote the point that lies on the segment joining the origin to

P

and such that the distance

P P'

is equal to 1 \slash 100. As

P

I

P'

describes a curve

C

I

C

has greater length?

1

Putnam 1947 B2

f(x)

be a differentiable function defined on the interval

(0,1)

|f'(x)| \leq M

0<x<1

and a positive real number

M.

\left| \int_{0}^{1} f(x)\; dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.

1

Putnam 1947 B1

f(x)

be a function such that

f(1)=1

x \geq 1

f'(x)= \frac{1}{x^2 +f(x)^{2}}.

\lim_{x\to \infty} f(x)

exists and is less than

1+ \frac{\pi}{4}.

1

Putnam 1947 A6

3\times 3

matrix has determinant

0

and the cofactor of any element is equal to the square of that element. Show that every element in the matrix is

0.

1

Putnam 1947 A5

a_1 , b_1 , c_1

be positive real numbers whose sum is

1,

n=1, 2, \ldots

a_{n+1}= a_{n}^{2} +2 b_n c_n, \;\;\;b_{n+1}= b_{n}^{2} +2 a_n c_n, \;\;\; c_{n+1}= c_{n}^{2} +2 a_n b_n.

a_n , b_n ,c_n

approach limits as

n\to \infty

and find those limits.

1

Putnam 1947 A4

A coast artillery gun can fire at every angle of elevation between

0^{\circ}

90^{\circ}

in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant (

=v_0

), determine the set

H

of points in the plane and above the horizontal which can be hit.

1

Putnam 1947 A3

Given a triangle

ABC

with an interior point

P

Q_1 , Q_2

not lying on any of the segments

AB , AC ,BC,

AP ,BP ,CP,

show that there does not exist a polygonal line

K

Q_1

Q_2

K

crosses each segment exactly once, ii)

K

does not intersect itself iii)

K

does not pass through

A, B , C

P.

1

Putnam 1947 A2

A real valued continuous function

f

satisfies for all real

x

y

the functional equation

f(\sqrt{x^2 +y^2 })= f(x)f(y).

f(x) =f(1)^{x^{2}}.

1

Putnam 1947 A1

(a_n)

is a sequence of real numbers such that for

n \geq 1

(2-a_n )a_{n+1} =1,

\lim_{n\to \infty} a_n =1.