1941 Putnam

Part of Putnam

Subcontests

(14)

1

Putnam 1941 B7

Do either (1) or (2): (1) Show that any solution

f(t)

of the functional equation

f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1

x,y\in \mathbb{R}

f''(t)= \pm c^{2} f(t)

c

, assuming the existence and continuity of the second derivative. Deduce that

f(t)

is one of the functions

\pm \cos ct, \;\;\; \pm \cosh ct.

(a_{i})_{i=1,...,n}

(b_{i})_{i=1,...,n}

be real numbers. Define an

(n+1)\times (n+1)

A=(c_{ij})

c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.

p(x)

is defined by the equation

\det A=0

f

be a polynomial and replace

(b_{i})

(f(b_{i}))

\det A=0

defines another polynomial

q(x)

f(p(x))-q(x)

is a multiple of

\prod_{i=1}^{n} (x-a_{i}).

1

Putnam 1941 B6

f(x)

is continuous in the interval

(0,1)

\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z)\;dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t)\; dt\right)^{3}.

1

Putnam 1941 B5

A car is being driven so that its wheels, all of radius

a

feet, have an angular velocity of

\omega

radians per second. A particle is thrown off from the tire of one of these wheels, where it is supposed that

a \omega^{2} >g

. Neglecting the resistance of the air, show that the maximum height above the roadway which the particle can reach is

\frac{(a \omega+g \omega^{-1})^{2}}{2g}.

1

Putnam 1941 B4

Given two perpendicular diameters

AB

CD

of an ellipse, we say that the diameter

A'B'

is conjugate to

AB

A'B'

is parallel to the tangent to the ellipse at

A

A'B'

be conjugate to

AB

C'D'

be conjugate to

CD

. Prove that the rectangular hyperbola through

A', B', C'

D'

passes through the foci of the ellipse.

1

Putnam 1941 B3

y_1

y_2

be two linearly independent solutions of the equation

y''+P(x)y'+Q(x)=0.

Find the differential equation satisfied by the product

z=y_1 y_2

1

Putnam 1941 B2

\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i^{2}}}

\lim_{n\to \infty} \sum_{i=1}^{n} \frac{1}{\sqrt{n^2 +i}}

\lim_{n\to \infty} \sum_{i=1}^{n^{2}} \frac{1}{\sqrt{n^2 +i}}

1

Putnam 1941 B1

(x,y)

moves so that its angular velocities about

(1,0)

(-1,0)

are equal in magnitude but opposite in sign. Prove that

y(x^2 +y^2 +1)\; dx= x(x^2 +y^2 -1) \;dy,

and verify that this is the differential equation of the family of rectangular hyperbolas passing through

(1,0)

(-1,0)

and having the origin as center.

1

Putnam 1941 A7

Do either (1) or (2): (1) Prove that the determinant of the matrix

\begin{pmatrix} 1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\ 2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\ 2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2 \end{pmatrix}

(1+a^2 +b^2 +c^2)^{3}

.
(2) A solid is formed by rotating the first quadrant of the ellipse

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

x

-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if

\frac{a}{b}\leq \sqrt{\frac{8}{5}}

1

Putnam 1941 A6

x

\overline{x}

of the center of mass of the area lying between the

x

-axis and the curve

y=f(x)

f(x)>0

, and between the lines

x=0

x=a

\overline{x}=g(a),

f(x)=A\cdot \frac{g'(x)}{(x-g(x))^{2}} \cdot e^{\int \frac{1}{t-g(t)} dt},

A

is a positive constant.

1

Putnam 1941 A5

L

is parallel to the plane

y=z

and meets the parabola

2x=y^2 ,z=0

and the parabola

3x=z^2, y=0

. Prove that if

L

moves freely subject to these constraints then it generates the surface

x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)

1

Putnam 1941 A4

a,b,c

f(x)=x^3 +p x^2 + qx+r

be real, and let

a\leq b\leq c

f'(x)

has a root in the interval

\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]

. What will be the form of

f(x)

if the root in question falls at either end of the interval?

1

Putnam 1941 A3

A circle of radius

a

rolls in the plane along the

x

-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter

a

, likewise rolling in the plane along the

x

1

Putnam 1941 A2

n

-th derivative with respect to

x

\int_{0}^{x} \left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{2!}+\ldots+\frac{(x-t)^{n-1}}{(n-1)!}\right)e^{nt} dt.

1

Putnam 1941 A1

Prove that the polynomial

(a-x)^6 -3a(a-x)^5 +\frac{5}{2} a^2 (a-x)^4 -\frac{1}{2} a^4 (a-x)^2

takes only negative values for

0<x<a