1977 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1977 B4

C

be a continuous closed curve in the plane which does not cross itself and let

Q

be a point inside

C

. Show that there exists points

P_1

P_2

C

Q

is the midpoint of the line segment

P_1P_2.

1

Putnam 1977 B6

H

be a subgroup with

h

elements in a group

G.

G

a

such that for all

x

H,

(xa)^3=1,

the identity. In

G

P

be the subset of all products

x_1ax_2a\dots x_na,

n

a positive integer and the

x_i

H.

P

is a finite set. (b) Show that, in fact,

P

has no more that

3h^2

1

Putnam 1977 B5

a_1,a_2,\dots a_n

(n>1)

A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.

A<2a_ia_j

1\leq i<j\leq n.

1

Putnam 1977 B3

An (ordered) triple

(x_1,x_2,x_3)

of positive irrational numbers with

x_1+x_2+x_3=1

is called balanced if each

x_i< 1/2.

If a triple is not balanced, say if

x_j>1/2

, one performs the following balancing act

B(x_1,x_2,x_3)=(x'_1,x'_2,x'_3),

x'_i=2x_i

i\neq j

x'_j=2x_j-1.

If the new triple is not balanced, one performs the balancing act on it. Does the continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?

1

Putnam 1977 B2

Given a convex quadrilateral

ABCD

O

not in the plane

ABCD

A'

OA,

B'

OB

C'

OC,

D'

OD

A'B'C'D'

is a parallelogram.

1

Putnam 1977 B1

Evaluate the infinite product

\prod_{n=2}^{\infty} \frac{n^3-1}{n^3+1}.

1

Putnam 1977 A6

f(x,y)

be a continuous function on the square

S=\{(x,y):0\leq x\leq 1, 0\leq y\leq 1\}.

(a,b)

in the interior of

S

S_{(a,b)}

be the largest square that is contained in

S

, is centered at

(a,b)

, and has sides parallel to those of

S

. If the double integral

\int \int f(x,y) dx dy

is zero when taken over each square

S_{(a,b)}

f(x,y)

be identically zero on

S

1

Putnam 1977 A5

\binom{pa}{pb}=\binom{a}{b} (\text{mod } p)

for all integers

p,a,

b

p

p>0,

a>b>0.

1

Putnam 1977 A4

0<x<1,

\sum_{n=0}^{\infty} \frac{x^{2^n}}{1-x^{2^{n+1}}}

as a rational function of

x.

1

Putnam 1977 A3

u,f

g

be functions, defined for all real numbers

x

\frac{u(x+1)+u(x-1)}{2}=f(x) \text{ and } \frac{u(x+4)+u(x-4)}{2}=g(x).

u(x)

f

g

1

Putnam 1977 A2

Determine all solutions in real numbers

x,y,z,w

x+y+z=w, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{w}.

1

Putnam 1977 A1

Consider all lines which meet the graph of

y=2x^4+7x^3+3x-5

in four distinct points, say

(x_i,y_i), i=1,2,3,4.

\frac{x_1+x_2+x_3+x_4}{4}

is independent of the line and find its value.