Putnam 1939

Part of Putnam

Subcontests

(14)

1

Putnam 1939 B7

(1)

(2)

:

(1)

ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!}

a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.

(2)

O

\lambda

a positive real number,

C

ax^2 + by^2 + cx + dy + e = 0,

C\lambda

ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0.

P

and a non-zero real number

k,

define the transformation

D(P,k)

as follows. Take coordinates

(x',y')

P

as the origin. Then

D(P,k)

(x',y')

(kx',ky').

D(O,\lambda)

D(A,-\lambda)

C

C\lambda,

A

(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))})

. Comment on the case

\lambda = 1.

1

Putnam 1939 B6

(1)

(2)

:

(1)

f

is continuous on the closed interval

[a, b]

and twice differentiable on the open interval

(a, b).

x_0 \in (a, b),

prove that we can find

\xi \in (a, b)

\dfrac{ ( \dfrac{(f(x_0) - f(a))}{(x_0 - a)} - \dfrac{(f(b) - f(a))}{(b - a)} )}{(x_0 - b)} = \dfrac{f''(\xi)}{2}.

(2)

AB

CD

are identical uniform rods, each with mass

m

2a.

They are placed a distance

b

ABCD

is a rectangle. Calculate the gravitational attraction between them. What is the limiting value as a tends to zero?

1

Putnam 1939 B5

(1)

(2)

:

(1)

\int_{1}^{k} [x] f'(x) dx = [k] f(k) - \sum_{1}{[k]} f(n),

k > 1,

[z]

denotes the greatest integer

\leq z.

Find a similar expression for:

\int_{1}^{k} [x^2] f'(x) dx.

(2)

A particle moves freely in a straight line except for a resistive force proportional to its speed. Its speed falls from

1,000 \dfrac{ft}{s}

900 \dfrac{ft}{s}

1200 ft.

Find the time taken to the nearest

0.01 s.

[No calculators or log tables allowed!]

1

Putnam 1939 B4

The axis of a parabola is its axis of symmetry and its vertex is its point of intersection with its axis. Find: the equation of the parabola which touches

y = 0

(1,0)

x = 0

(0,2);

the equation of its axis; and its vertex.

1

Putnam 1939 B3

a_n = (n^2 + 1) 3^n,

find a recurrence relation

a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.

\sum_{n\geq0} a_n x^n.

1

Putnam 1939 B2

\int_{1}^{3} ( (x - 1)(3 - x) )^{\dfrac{-1}{2}} dx

\int_{1}^{\infty} (e^{x+1} + e^{3-x})^{-1} dx.

1

Putnam 1939 B1

P(a,b)

Q(0,c)

are on the curve

\dfrac{y}{c} = \cosh{(\dfrac{x}{c})}.

The line through

Q

parallel to the normal at

P

x-

R.

QR = b.

1

Putnam 1939 A7

(1)

(2)

:

(1)

C_a

(y - a^2)^2 = x^2(a^2 - x^2).

Find the curve which touches all

C_a

a > 0.

Sketch the solution and at least two of the

C_a.

(2)

(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,

(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.

1

Putnam 1939 A6

(1)

(2)

:

(1)

A circle radius

r

rolls around the inside of a circle radius

3r,

so that a point on its circumference traces out a curvilinear triangle. Find the area inside this figure.

(2)

A frictionless shell is fired from the ground with speed

v

at an unknown angle to the vertical. It hits a plane at a height

h.

Show that the gun must be sited within a radius

\frac{v}{g} (v^2 - 2gh)^{\frac{1}{2}}

of the point directly below the point of impact.

1

Putnam 1939 A5

(1)

(2)

(1)

x

y

are functions of

t.

x' = x + y - 3, y' = -2x + 3y + 1,

x(0) = y(0) = 0.

(2)

A weightless rod is hinged at

O

so that it can rotate without friction in a vertical plane. A mass

m

is attached to the end of the rod

A,

which is balanced vertically above

O.

t = 0,

the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under

O

t = \sqrt{(\frac{OA}{g})} \ln{(1 + sqrt(2))}.

1

Putnam 1939 A4

4

lines in Euclidean

3-

L_1: x = 1, y = 0;

L_2: y = 1, z = 0;

L_3: x = 0, z = 1;

L_4: x = y, y = -6z.

Find the equations of the two lines which both meet all of the

L_i.

1

Putnam 1939 A3

x^3 + a x^2 + b x + c = 0

\alpha, \beta

\gamma.

Find the cubic whose roots are

\alpha^3, \beta^3, \gamma^3.

1

Putnam 1939 A2

C

y = x^3

x

takes all real values). The tangent at

A

meets the curve again at

B.

Prove that the gradient at

B

4

times the gradient at

A.

1

Putnam 1939 A1

C

y^2 = x^3

x

takes all non-negative real values). Let

O

be the origin, and

A

be the point where the gradient is

1.

Find the length of the curve from

O

A.