1960 Putnam

Part of Putnam

Subcontests

(14)

1

Putnam 1960 B7

g(t)

h(t)

be real, continuous functions for

t\geq 0.

Show that any function

v(t)

satisfying the differential inequality

\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,

satisfies the further inequality

v(t)\geq u(t),

\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.

From this, conclude that for sufficiently small

t>0,

the solution of

\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c

v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),

where the maximum is over all continuous functions

w(t)

defined over some

t

[0,t_0 ].

1

Putnam 1960 B1

Find all integer solutions

(m,n)

m^{n}=n^{m}.

1

Putnam 1960 B6

Any positive integer

n

can be written in the form

n=2^{k}(2l+1)

k,l

positive integers. Let

a_n =e^{-k}

b_n = a_1 a_2 a_3 \cdots a_n.

\sum_{n=1}^{\infty} b_n

1

Putnam 1960 B5

Define a sequence

(a_n)

a_0 =0

a_n = 1 +\sin(a_{n-1}-1)

n\geq 1

\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^{n} a_k.

1

Putnam 1960 B4

Consider the arithmetic progression

a, a+d, a+2d,\ldots

a

d

are positive integers. For any positive integer

k

, prove that the progression has either no

k

-th powers or infinitely many.

1

Putnam 1960 B3

The motion of the particles of a fluid in the plane is specified by the following components of velocity

\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.

Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as

t\to \infty

, and justify your conclusion.

1

Putnam 1960 B2

Evaluate the double series

\sum_{j=0}^{\infty} \sum_{k=0}^{\infty} 2^{-3k -j -(k+j)^{2}}.

1

Putnam 1960 A7

N(n)

denote the smallest positive integer

N

x^N =e

for every element

x

of the symmetric group

S_n

e

denotes the identity permutation. Prove that if

n>1,

\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\ 1\; \text{otherwise}. \end{cases}

1

Putnam 1960 A6

A player repeatedly throwing a die is to play until their score reaches or passes a total

n

p(n)

the probability of making exactly the total

n,

and find the value of

\lim_{n \to \infty} p(n).

1

Putnam 1960 A5

Find all polynomials

f(x)

with real coefficients having the property

f(g(x))=g(f(x))

for every polynomial

g(x)

with real coefficients.

1

Putnam 1960 A4

Given two points,

P

Q

, on the same side of a line

L

, the problem is to find a third point

R

PR+ RQ+RS

is minimal, where

S

is the unique point on

L

RS

is perpendicular to

L.

Consider all cases.

1

Putnam 1960 A3

t_1 , t_2, t_3, t_4, t_5

are real numbers, then

\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.

1

Putnam 1960 A2

Show that if three points are inside are closed square of unit side, then some pair of them are within

\sqrt{6}-\sqrt{2}

1

Putnam 1960 A1

n

be a given positive integer. How many solutions are there in ordered positive integer pairs

(x,y)

to the equation

\frac{xy}{x+y}=n?