MathDB

1

Putnam 1958 February B7

f(x)

is continuous for

a\leq x \leq b

and

\int_{a}^{b} x^n f(x) \, dx =0

for

n=0,1,2, \ldots,

then

f(x)

is identically zero on

a \leq x \leq b.

B6

1

Putnam 1958 February B6

A projectile moves in a resisting medium. The resisting force is a function of the velocity and is directed along the velocity vector. The equation

x=f(t)

(where

f(t)

is not constant) gives the horizontal distance in terms of the time

t

. Show that the vertical distance

y

is given by

y=-gf(t) \int \frac{dt}{f'(t)} + g \int \frac{f(t)}{f'(t)} \, dt +Af(t)+B

where

A

and

B

are constants and

g

is the acceleration due to gravity.

B3

1

Putnam 1958 February B3

In a round-robin tournament with

n

players in which there are no draws, the numbers of wins scored by the players are

s_1 , s_2 , \ldots, s_n

. Prove that a necessary and sufficient condition for the existence of three players

A,B,C

such that

A

beats

B

,

B

beats

C

, and

C

beats

A

is

s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.

B2

1

Putnam 1958 February B2

Prove that the product of four consecutive positive integers cannot be a perfect square or cube.

B1

1

Putnam 1958 February B1

i) Given line segments

A,B,C,D

with

A

the longest, construct a quadrilateral with these sides and with

A

and

B

parallel, when possible. ii) Given any acute-angled triangle

ABC

and one altitude

AH

, select any point

D

on

AH

, then draw

BD

and extend until it intersects

AC

in

E

, and draw

CD

and extend until it intersects

AB

in

F

. Prove that

\angle AHE = \angle AHF

.

A7

1

Putnam 1958 February A7

Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others.

A6

1

Putnam 1958 February A6

What is the smallest amount that may be invested at interest rate

i

, compounded annually, in order that one may withdraw

1

dollar at the end of the first year,

4

dollars at the end of the second year,

\ldots

,

n^2

dollars at the end of the

n

-th year, in perpetuity?

A5

1

Putnam 1958 February A5

Show that the integral equation

f(x,y) = 1 + \int_{0}^{x} \int_{0}^{y} f(u,v) \, du \, dv

has at most one solution continuous for

0\leq x \leq 1, 0\leq y \leq 1.

A4

1

Putnam 1958 February A4

If

a_1 ,a_2 ,\ldots, a_n

are complex numbers such that

|a_1| =|a_2 | =\cdots = |a_n| =r \ne 0,

and if

T_s

denotes the sum of all products of these

n

numbers taken

s

at a time, prove that

\left| \frac{T_s }{T_{n-s}}\right| =r^{2s-n}

whenever the denominator of the left-hand side is different from

0

.

A3

1

Putnam 1958 February A3

Real numbers are chosen at random from the interval

[0,1].

If after choosing the

n

-th number the sum of the numbers so chosen first exceeds

1

, show that the expected value for

n

is

e

.

A2

1

Putnam 1958 February A2

Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.

A1

1

Putnam 1958 February A1

If

a_0 , a_1 ,\ldots, a_n

are real number satisfying

\frac{a_0 }{1} + \frac{a_1 }{2} + \ldots + \frac{a_n }{n+1}=0,

show that the equation

a_n x^n + \ldots +a_1 x+a_0 =0

has at least one real root.

B4

1

Average distance between two points on unit sphere

Title is self explanatory. Pick two points on the unit sphere. What is the expected distance between them?

B5

1

Prove that S is a subset of a straight line ...

S

is an infinite set of points in the plane. The distance between any two points of

S

is integral. Prove that

S

is a subset of a straight line.

1958 February Putnam

Subcontests

Putnam 1958 February B7

Putnam 1958 February B6

Putnam 1958 February B3

Putnam 1958 February B2

Putnam 1958 February B1

Putnam 1958 February A7

Putnam 1958 February A6

Putnam 1958 February A5

Putnam 1958 February A4

Putnam 1958 February A3

Putnam 1958 February A2

Putnam 1958 February A1

Average distance between two points on unit sphere

Prove that S is a subset of a straight line ...