1946 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1946 B2

A, B

be two variable points on a parabola

P_{0}

, such that the tangents at

A

B

are perpendicular to each other. Show that the locus of the centroid of the triangle formed by

A,B

and the vertex of

P_0

P_1 .

Apply the same process to

P_1

and repeat the process, obtaining the sequence of parabolas

P_1, P_2 , \ldots, P_n

. If the equation of

P_0

y=m x^2

, find the equation of

P_n .

1

Putnam 1946 B1

K

denote the circumference of a circular disk of radius

1

k

denote a circular arc that joins two points

a,b

K

and lies otherwise in the given circular disc. Suppose that

k

divides the circular disk into two parts of equal area. Prove that the length of

k

2.

1

Putnam 1946 B5

\lceil (\sqrt{3}+1)^{2n})\rceil

is divisible by

2^{n+1}.

1

Putnam 1946 B3

In a solid sphere of radius

R

\rho

is a function of

r

, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is

k r^2

k

is a constant, find

\rho

as a function of

r.

Find also the magnitude of the force of attraction at a point outside the sphere at a distance

r

from the center.

1

Putnam 1946 B6

A particle moves on a circle with center

O

, starting from rest at a point

P

and coming to rest again at a point

Q

, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between

P

Q

and that, at some point

R

P

Q

, the acceleration vector points in along the radius

RO.

1

Putnam 1946 B4

For each positive integer

n

p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.

h_1 < h_2 < h_3 <\ldots

1

Putnam 1946 A6

A particle of unit mass moves on a straight line under the action of a force which is a function

f(v)

of the velocity

v

of the particle, but the form of the function is not known. A motion is observed, and the distance

x

covered in time

t

satisfies the formula

x= at^2 + bt+c

a,b,c

have numerical values determined by observation of the motion. Find the function

f(v)

for the range of

v

covered by the experiment.

1

Putnam 1946 A5

Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1.

1

Putnam 1946 A4

g(x)

be a function that has a continuous first derivative

g'(x)

g(0)=0

|g'(x)| \leq |g(x)|

for all values of

x.

g(x)

vanishes identically.

1

Putnam 1946 A3

A projectile in flight is observed simultaneously from four radio stations which are situated at the corners of a square of side

b

. The distances of the projectile from the four stations, taken in order around the square, are found to be

R_1 , R_2 , R_3

R_4

R_{1}^{2}+ R_{3}^{2}= R_{2}^{2}+ R_{4}^{2}.

Show also that the height

h

of the projectile above the ground is given by

h^{2}=- \frac{1}{2} b^2 +\frac{1}{4}(R_{1}^{2}+R_{2}^{2}+R_{3}^{2}+R_{4}^{2}) -\frac{1}{8 b^{2}}(R_{1}^{4}+R_{2}^{4}+R_{3}^{4}+R_{4}^{4}- 2 R_{1}^{2}R_{3}^{2} -2 R_{2}^{2} R_{4}^{2}).

1

Putnam 1946 A2

a(x), b(x), c(x)

d(x)

are polynomials in

x

\int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx

is divisible by

(x-1)^4.

1

Putnam 1946 A1

Suppose that the function

f(x)=a x^2 +bx+c

a,b,c

are real, satisfies the condition

|f(x)|\leq 1

|x|\leq1

|f'(x)|\leq 4

|x|\leq1