1961 Putnam

Part of Putnam

Subcontests

(14)

1

Putnam 1961 B7

Given a sequence

(a_n)

of non-negative real numbers such that

a_{n+m}\leq a_{n} a_{m}

for all pairs of positive integers

m

n,

prove that the sequence

(\sqrt[n]{a_n })

1

Putnam 1961 B6

Consider the function

y(x)

satisfying the differential equation

y'' = -(1+\sqrt{x})y

y(0)=1

y'(0)=0.

y(x)

vanishes exactly once on the interval 0< x< \pi \slash 2, and find a positive lower bound for the zero.

1

Putnam 1961 B5

k

be a positive integer, and

n

a positive integer greater than

2

f_{1}(n)=n,\;\; f_{2}(n)=n^{f_{1}(n)},\;\ldots\;, f_{j+1}(n)=n^{f_{j}(n)}.

Prove either part of the inequality

f_{k}(n) < n!! \cdots ! < f_{k+1}(n),

where the middle term has

k

factorial symbols.

1

Putnam 1961 B4

x_1 , x_2 ,\ldots, x_n

be real numbers in

[0,1].

Determine the maximum value of the sum of the

\frac{n(n-1)}{2}

\sum_{i<j}|x_i -x_j |.

1

Putnam 1961 B3

Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.

1

Putnam 1961 B2

a

b

be given positive real numbers, with

a<b.

If two points are selected at random from a straight line segment of length

b,

what is the probability that the distance between them is at least

a?

1

Putnam 1961 B1

a_1 , a_2 , a_3 ,\ldots

be a sequence of positive real numbers, define

s_n = \frac{a_1 +a_2 +\ldots+a_n }{n}

r_n = \frac{a_{1}^{-1} +a_{2}^{-1} +\ldots+a_{n}^{-1} }{n}.

\lim_{n\to \infty} s_n

\lim_{n\to \infty} r_n

exist, prove that the product of these limits is not less than

1.

1

Putnam 1961 A7

S

be a nonempty closed set in the euclidean plane for which there is a closed disk

D

S

D

is a subset of every closed disk that contains

S

. Prove that every point inside

D

is the midpoint of a segment joining two points of

S.

1

Putnam 1961 A6

p(x)=1+x+x^2 +\ldots+x^n

is reducible over

\mathbb{F}_{2}

n+1

is composite. If

n+1

p(x)

irreducible over

\mathbb{F}_{2}

1

Putnam 1961 A5

\Omega

n

n>2

\Sigma

be a nonempty subcollection of the

2^n

\Omega

that is closed with respect to the unions, intersections and complements. If

k

is the number of elements of

\Sigma,

what are the possible values of

k?

1

Putnam 1961 A4

\Omega(n)

be the number of prime factors of

n

f(1)=1

f(n)=(-1)^{\Omega(n)}.

Furthermore, let

F(n)=\sum_{d|n} f(d).

F(n)=0,1

for all positive integers

n

. For which integers

n

F(n)=1?

1

Putnam 1961 A3

\lim_{n\to \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^2 +j^2 }.

1

Putnam 1961 A2

For a real-valued function

f(x,y)

of two positive real variables

x

y

f

to be linearly bounded if and only if there exists a positive number

K

|f(x,y)| < K(x+y)

for all positive

x

y.

Find necessary and sufficient conditions on the real numbers

\alpha

\beta

x^{\alpha}y^{\beta}

is linearly bounded.

1

Putnam 1961 A1

The graph of the equation

x^y =y^x

in the first quadrant consists of a straight line and a curve. Find the coordinates of the intersection of the line and the curve.