1942 Putnam

Part of Putnam

Subcontests

(11)

1

Putnam 1942 B5

Sketch the curve

y= \frac{x}{1+x^6 (\sin x)^{2}},

\int_{0}^{\infty} \frac{x}{1+x^6 (\sin x)^{2}}\; dx

1

Putnam 1942 B4

A particle moves under a central force inversely proportional to the

k

-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find

k

1

Putnam 1942 B3

x=\phi(u,v)

y=\psi(u,v)

\phi

\psi

are solutions of the partial differential equation

(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.

By assuming that

x

y

are the independent variables, show that

(1)

may be transformed to

(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.

(2)

and show how this effects in general the solution of

(1)

. What other solutions does

(1)

1

Putnam 1942 B2

For the family of parabolas

y= \frac{ a^3 x^{2}}{3}+ \frac{ a^2 x}{2}-2a

(i) find the locus of vertices, (ii) find the envelope, (iii) sketch the envelope and two typical curves of the family.

1

Putnam 1942 B1

A square of side

2a

, lying always in the first quadrant of the

xy

-plane, moves so that two consecutive vertices are always on the

x

y

-axes respectively. Prove that a point within or on the boundary of the square will in general describe a portion of a conic. For what points of the square does this locus degenerate?

1

Putnam 1942 A6

Any circle in the

xy

-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid. By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.

1

Putnam 1942 A5

A circle of radius

a

is revolved through

180^{\circ}

about a line in its plane, distant

b

from the center of the circle, where

b>a

. For what value of the ratio

\frac{b}{a}

does the center of gravity of the solid thus generated lie on the surface of the solid?

1

Putnam 1942 A4

Find the orthogonal trajectories of the family of conics

(x+2y)^{2} = a(x+y)

. At what angle do the curves of one family cut the curves of the other family at the origin?

1

Putnam 1942 A3

\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}

convergent or divergent?

1

Putnam 1942 A2

If a polynomial

f(x)

(x-a)^{2} (x-b)

a\ne b

, derive a formula for the remainder.

1

Putnam 1942 A1

A square of side

2a

, lying always in the first quadrant of the

xy

-plane, moves so that two consecutive vertices are always on the

x

y

-axes respectively. Find the locus of the midpoint of the square.