1954 Putnam

Part of Putnam

Subcontests

(14)

1

Putnam 1954 B7

a>0

\lim_{n \to \infty} \sum_{s=1}^{n} \left( \frac{a+s}{n} \right)^{n}

e^a

e^{a+1}.

1

Putnam 1954 B5

f(x)

be a real-valued function, defined for

-1<x<1

f'(0)

(a_n) , (b_n)

be two sequences such that

-1 <a_n <0 <b_n <1

n

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

\lim_{n \to \infty} \frac{ f(b_n )- f(a_n ) }{b_n -a_n} =f'(0).

1

Putnam 1954 B4

Given the focus

F

and the directrix

D

P

L

, describe a euclidean construction for the point or points of intersection of

P

L.

Be sure to identify the case for which there are no points of intersection.

1

Putnam 1954 B3

[a_1 , b_1 ] , \ldots, [a_n ,b_n ]

be a collection of closed intervals such that any of these closed intervals have a point in common. Prove that there exists a point contained in every one of these intervals.

1

Putnnam 1954 B2

s

denote the sum of the alternating harmonic series. Rearrange this series as follows

1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +\frac{1}{7} - \frac{1}{4} + \frac{1}{9} + \frac{1}{11} - \ldots

Assume as known that this series converges as well and denote its sum by

S

s_k, S_k

respectively the

k

-th partial sums of both series. Prove that

\!\!\!\! \text{i})\; S_{3n} = s_{4n} +\frac{1}{2} s_{2n}.

\text{ii}) \; S\ne s.

1

Putnam 1954 B1

Show that the equation

x^2 -y^2 =a^3

has always integral solutions for

x

y

a

is a positive integer.

1

Putnam 1954 A7

Prove that there are no integers

x

y

x^2 +3xy-2y^2 =122.

1

Putnam 1954 A6

u_0 , u_1 ,\ldots

is a sequence of real numbers such that

u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots

\sum u_n

converges, then

u_k=0

k

1

Putnam 1954 A5

f(x)

be a real-valued function defined for

0<x<1.

\lim_{x \to 0} f(x) =0 \;\; \text{and} \;\; f(x) - f \left( \frac{x}{2} \right) =o(x),

f(x) =o(x),

where we use the O-notation.

1

Putnam 1954 A4

A uniform rod of length

2k

w

rests with the end

A

against a vertical wall, while the lower end

B

is fastened by a string

BC

2b

coming from a point

C

in the wall above

A.

If the system is in equilibrium, determine the angle

ABC.

1

Putnam 1954 A3

Prove that if the family of integral curves of the differential equation

\frac{dy}{dx} +p(x) y= q(x),

p(x) q(x) \ne 0

, is cut by the line

x=k

the tangents at the points of intersection are concurrent.

1

Putnam 1954 A2

Consider any five points in the interior of square

S

1

. Prove that at least one of the distances between these points is less than \sqrt{2} \slash 2. Can this constant be replaced by a smaller number?

1

Putnam 1954 A1

n

be an odd integer greater than

1.

A

n\times n

symmetric matrix such that each row and column consists of some permutation of the integers

1,2, \ldots, n.

Show that each of the integers

1,2, \ldots, n

must appear in the main diagonal of

A

1

Problem about there exits

x \in \mathbb{Q}^+

. Prove that there exits

\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}

and pairwe distinct such that

x= \sum_{i=1}^{k} \frac{1}{\alpha_i}