1979 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1979 B4

(a) Find a solution that is not identically zero, of the homogeneous linear differential equation

(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=0.

Intelligent guessing of the form of a solution may be helpful.
(b) Let

y=f(x)

be the solution of the nonhomogeneous differential equation

(3x^2-x-1)y''-(9x^2+9x-2)y'+(18x+3)y=6(6x+1)

f(0)=1

(f(-1)-2)(f(1)-6)=1.

a,b,c

(f(-2)-a)(f(2)-b)=c.

1

Putnam 1979 B6

k=1,2 \dots, n

z_k=x_k+iy_k,

x_k

y_k

i=\sqrt{-1}

r

bet the absolute value of the real part of

\pm \sqrt{z_1^2+z_2^2+\dots z_n^2}.

r\leq |x_1|+|x_2|+ \dots +|x_n|.

1

Putnam 1979 B5

In the plane, let

C

be a closed convex set that contains

(0,0)

but no other point with integer coordinates. Suppose that

A(C)

C

, is equally distributed among the four quadrants. Prove that

A(C) \leq 4.

1

Putnam 1979 B3

F

be a finite field having an odd number

m

of elements. Let

p(x)

be an irreducible (i.e. nonfactorable) polynomial over

F

x^2+bx+c, ~~~~~~ b,c \in F.

For how many elements

k

F

p(x)+k

irreducible over

F

1

Putnam 1979 B2

0<a<b.

\lim_{t\to 0} \{ \int_{0}^{1} [bx+a(1-x)]^t dx\}^{1/t}.

[The final answer should not involve any other operations other than addition, subtraction, multiplication, division and exponentiation.]

1

Putnam 1979 B1

Prove or disprove: there is at least one straight line normal to the graph of

y=\cosh x

(a,\cosh a)

and also normal to the graph of

y=

\sinh x

(c,\sinh c).

1

Putnam 1979 A6

0\leq p_i \leq 1

i=1,2, \dots, n.

\sum_{i=1}^{n} \frac{1}{|x-p_i|} \leq 8n(1+1/3+1/5+\dots +\frac{1}{2n-1})

x

0\leq x \leq 1.

1

Putnam 1979 A5

\lceil x \rceil

the greatest integer less than or equal to

x

S(x)

\lceil x \rceil, \lceil 2x \rceil, \lceil 3x \rceil, \dots.

Prove that there are distinct real solutions

\alpha

\beta

of the equation

x^3-10x^2+29x-25=0

such that infinitely many positive integers appear both in

S(\alpha)

S(\beta).

1

Putnam 1979 A4

A

2n

points in the plane, no three of which are collinear. Suppose that

n

of them are colored red and the remaining

n

blue. Prove or disprove: there are

n

closed straight line segments, no two with a point in common, such that the endpoints of each segment are points of

A

having different colors.

1

Putnam 1979 A3

x_1,x_2,x_3, \dots

be a sequence of nonzero real numbers satisfying

x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.

Establish necessary and sufficient conditions on

x_1

x_2

x_n

to be an integer for infinitely many values of

n.

1

Putnam 1979 A2

Establish necessary and sufficient conditions on the constant

k

for the existence of a continuous real valued function

f(x)

f(f(x))=kx^9

x

1

Putnam 1979 A1

Find positive integers

n

a_1, a_2, \dots, a_n

a_1+a_2+\dots a_n=1979

and the product

a_1a_2\dots a_n

as large as possible.