1993 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1993 B6

S

be a set of three, not necessarily distinct, positive integers. Show that one can transform

S

into a set containing

0

by a finite number of applications of the following rule: Select two of the integers

x

y

x\leq y

and replace them with

2x

y-x.

1

Putnam 1993 B5

Show that given any

4

points in the plane we can find two whose distance apart is not an odd integer.

1

Putnam 1993 B4

K(x, y), f(x)

g(x)

are positive and continuous for

x, y \subseteq [0, 1]

\int_{0}^{1} f(y) K(x, y) dy = g(x)

\int_{0}^{1} g(y) K(x, y) dy = f(x)

x \subseteq [0, 1]

f = g

[0, 1]

1

Putnam 1993 B3

x

y

are chosen at random (with uniform density) from the interval

(0, 1)

. What is the probability that the closest integer to

x/y

1

Putnam 1993 B2

2n

cards numbered from

1

2n

is shuffled and n cards are dealt to

A

B

A

B

alternately discard a card face up, starting with

A

. The game when the sum of the discards is first divisible by

2n + 1

, and the last person to discard wins. What is the probability that

A

wins if neither player makes a mistake?

1

Putnam 1993 B1

What is the smallest integer

n > 0

such that for any integer m in the range

1, 2, 3, ... , 1992

we can always find an integral multiple of

\frac{1}{n}

in the open interval

(\frac{m}{1993}, \frac{m+1}{1994})

1

Putnam 1993 A6

a_0, a_1, a_2, ...

be a sequence such that:

a_0 = 2

a_n = 2

3; a_n =

3

n

n+1

2

in the sequence. So the sequence starts:

233233323332332 ...

. Show that we can find

\alpha

a_n = 2

n = [\alpha m]

for some integer

m \geq 0

1

Putnam 1993 A5

Let U be the set formed as the union of three open intervals,

U = (-100, -10) \cup (1/101, 1/11) \cup (101/100, 11/10)

\int_{U} \frac{(x^2-x)^2}{(x^3-3x+1)^2} dx

1

Putnam 1993 A4

Given a sequence of

19

positive (not necessarily distinct) integers not greater than

93

93

positive (not necessarily distinct) integers not greater than

19

. Show that we can find non-empty subsequences of the two sequences with equal sum.

1

Putnam 1993 A3

P

be the set of all subsets of

{1, 2, ... , n}

. Show that there are

1^n + 2^n + ... + m^n

f : P \longmapsto {1, 2, ... , m}

f(A \cap B) = min( f(A), f(B))

A, B.

1

Putnam 1993 A2

The sequence an of non-zero reals satisfies

a_n^2 - a_{n-1}a_{n+1} = 1

n \geq 1

. Prove that there exists a real number

\alpha

a_{n+1} = \alpha a_n - a_{n-1}

n \geq 1

1

Putnam 1993 A1

O

y = c

intersects the curve

y = 2x - 3x^3

P

Q

in the first quadrant and cuts the y-axis at

R

c

so that the region

OPR

bounded by the y-axis, the line

y = c

and the curve has the same area as the region between

P

Q

under the curve and above the line

y = c