Putnam 1938

Part of Putnam

Subcontests

(13)

1

Putnam 1938 B6

What is the shortest distance between the plane

Ax + By + Cz + 1 = 0

and the ellipsoid

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

You may find it convenient to use the notation

h = (A^2 + B^2 + C^2)^{\frac{-1}{2}}, m = (a^2A^2 + b^2B^2 + c^2C^2)^{\frac{1}{2}}.

What is the algebraic condition for the plane not to intersect the ellipsoid?

1

Putnam 1938 B5

Find the locus of the foot of the perpendicular from the center of a rectangular hyperbola to a tangent. Obtain its equation in polar coordinates and sketch it.

1

Putnam 1938 B4

P

has focus a distance

m

from the directrix. The chord

AB

P

A.

What is the minimum length for

AB?

1

Putnam 1938 B3

A horizontal disk diameter

3

inches rotates once every

15

seconds. An insect starts at the southernmost point of the disk facing due north. Always facing due north, it crawls over the disk at

1

inch per second. Where does it again reach the edge of the disk?

1

Putnam 1938 B2

Find all solutions of the differential equation

zz" - 2z'z' = 0

which pass through the point

x=1, z=1.

1

Putnam 1938 B1

(1)

(2)

(1)

A

(a_{ij}), 1 \leq i,j \leq 4.

d =

(A),

A_{ij}

be the cofactor of

a_{ij}

, that is, the determinant of the

3 \times 3

matrix formed from

A

a_{ij}

and other elements in the same row and column. Let

B

4 \times 4

(A_{ij})

D

B.

D = d^3

.

(2)

P(x)

be the quadratic

Ax^2 + Bx + C.

P(x) = x

has unequal real roots. Show that the roots are also roots of

P(P(x)) = x.

Find a quadratic equation for the other two roots of this equation. Hence solve

(y^2 - 3y + 2)2 - 3(y^2 - 3y + 2) + 2 - y = 0.

1

Putnam 1938 A7

(1)

(2)

(1)

S

is a thin spherical shell of constant thickness and density with total mass

M

O.

P

is a point outside

S.

Prove that the gravitational attraction of

S

P

is the same as the gravitational attraction of a point mass

M

O.

(2)

K

z = xy

3-

space. Find all straight lines lying in

S

. Draw a diagram to illustrate them.

1

Putnam 1938 A6

A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?

1

Putnam 1938 A5

(1)

\lim_{x \to \infty} \frac{x^2}{e^x}

(2)

\lim_{k \to 0} \frac{1}{k} \int_{0}^{k} (1 + \sin 2x)^{\frac{1}{x}} dx

1

Putnam 1938 A4

A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes. Each half-plane is bounded by a horizontal line passing through the axis of the cylinder. The angle between the two half-planes is

\theta

. Prove that the volume of the notch is minimized (for given tree and

\theta

) by taking the bounding planes at equal angles to the horizontal plane.

1

Putnam 1938 A3

A particle moves in the Euclidean plane. At time

t

(taking all real values) its coordinates are

x = t^3 - t

y = t^4 + t.

Show that its velocity has a maximum at

t = 0,

and that its path has an inflection at

t = 0.

1

Putnam 1938 A2

A solid has a cylindrical middle with a conical cap at each end. The height of each cap equals the length of the middle. For a given surface area, what shape maximizes the volume?

1

Putnam 1938 A1

A solid in Euclidean

3

-space extends from

z = \frac{-h}{2}

z = \frac{+h}{2}

and the area of the section

z = k

is a polynomial in

k

of degree at most

3

. Show that the volume of the solid is

\frac{h(B + 4M + T)}{6},

B

is the area of the bottom

(z = \frac{-h}{2})

M

is the area of the middle section

(z = 0),

T

is the area of the top

(z = \frac{h}{2})

. Derive the formulae for the volumes of a cone and a sphere.