1963 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1963 B6

E

be a Euclidean space of at most three dimensions. If

A

is a nonempty subset of

E

S(A)

to be the set of points that lie on closed segments joining pairs of points of

A

(a one-point set should be considered to be a special case of a closed segment). For a given nonempty set

A_0

A_n =S(A_{n-1})

n=1,2,\ldots

A_2 =A_3 =\ldots.

1

Putnam 1963 B5

(a_n )

be a sequence of real numbers satisfying the inequalities

0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,

and such that the series

\sum_{n=0}^{\infty} a_n

converges. Prove that

\lim_{n\to \infty} n a_n = 0.

1

Putnam 1963 B4

C

be a closed plane curve that has a continuously turning tangent and bounds a convex region. If

T

is a triangle inscribed in

C

with maximum perimeter, show that the normal to

C

at each vertex of

T

bisects the angle of

T

at that vertex. If a triangle

T

has the property just described, does it necessarily have maximum perimeter? What is the situation if

C

1

Putnam 1963 B3

Find every twice-differentiable function

f: \mathbb{R} \rightarrow \mathbb{R}

that satisfies the functional equation

f(x)^2 -f(y)^2 =f(x+y)f(x-y)

x,y \in \mathbb{R}.

1

Putnam 1963 B2

S

be the set of all numbers of the form

2^m 3^n

m

n

are integers. Is

S

dense in the set of positive real numbers?

1

Putnam 1963 B1

For what integers

a

x^2 -x+a

x^{13}+ x +90

1

Putnam 1963 A6

U

V

be any two distinct points on an ellipse, let

M

be the midpoint of the chord

UV

AB

CD

be any two other chords through

M

UV

AC

P

BD

Q

M

is the midpoint of the segment

PQ.

1

Putnam 1963 A5

i) Prove that if a function

f

is continuous on the closed interval

[0, \pi]

\int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,

then there exist points

0 < \alpha < \beta < \pi

f(\alpha) =f(\beta) =0.

R

be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of

R

bisects at least three different chords of the boundary of

R.

1

Putnam 1963 A4

(a_n)

be a sequence of positive real numbers. Show that

\limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1

1

cannot be replaced by any larger number.

1

Putnam 1963 A3

Find an integral formula for the solution of the differential equation

\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,

y

as a function of

f

satisfying the initial conditions

y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0

f

is continuous and

\delta

is the differential operator

x \frac{d}{dx}.

1

Putnam 1963 A2

f:\mathbb{N}\rightarrow \mathbb{N}

be a strictly increasing function such that

f(2)=2

f(mn)=f(m)f(n)

for every pair of relatively prime positive integers

m

n

f(n)=n

for every positive integer

n

1

Putnam 1963 A1

i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon. ii) Let

P_1 , P_2 ,\ldots, P_{12}

be the vertices of a regular dodecagon. Prove that the three diagonals

P_{1}P_{9}, P_{2}P_{11}

P_{4}P_{12}