MathDB

1

log sequence converges, limit is continuous

x>e^e

define a sequence

S_x=u_0,u_1,\ldots

recursively as follows:

u_0=e

, and for

n\ge0

,

u_{n+1}=\log_{u_n}x

. Prove that

S_x

converges to a number

g(x)

and that the function

g

defined in this way is continuous for

x>e^e

.

B4

1

n_1n_2... divides (n_1+k)(n_2+k)...

Let

n_1,n_2,\ldots,n_s

be distinct integers such that

(n_1+k)(n_2+k)\cdots(n_s+k)

is an integral multiple of

n_1n_2\cdots n_s

for every integer

k

. For each of the following assertions give a proof or a counterexample:

(\text a)

|n_i|=1

for some

i

(\text b)

If further all

n_i

are positive, then

\{n_1,n_2,\ldots,n_2\}=\{1,2,\ldots,s\}.

B3

1

probability c+d square, find limit

Let

p_n

be the probability that

c+d

is a perfect square when the integers

c

and

d

are selected independently at random from the set

\{1,2,\ldots,n\}

. Show that

\lim_{n\to\infty}p_n\sqrt n

exists and express this limit in the form

r(\sqrt s-t)

, where

s

and

t

are integers and

r

is a rational number.

B2

1

double integral with exp is polynomial

Let

A(x,y)

be the number of points

(m,n)

in the plane with integer coordinates

m

and

n

satisfying

m^2+n^2\le x^2+y^2

. Let

g=\sum_{k=1}^\infty e^{-k^2}

. Express

\int^\infty_{-\infty}\int^\infty_{-\infty}A(x,y)e^{-x^2-y^2}dxdy

as a polynomial in

g

.

B1

1

triangulating triangle and reassembling them

Let

M

be the midpoint of side

BC

of

\triangle ABC

. Using the smallest possible

n

, described a method for cutting

\triangle AMB

into

n

triangles which can be reassembled to form a triangle congruent to

\triangle AMC

.

A6

1

permutation of N, limit is 1

Let

\sigma

be a bijection on the positive integers. Let

x_1,x_2,x_3,\ldots

be a sequence of real numbers with the following three properties:

(\text i)

|x_n|

is a strictly decreasing function of

n

;

(\text{ii})

|\sigma(n)-n|\cdot|x_n|\to0

as

n\to\infty

;

(\text{iii})

\lim_{n\to\infty}\sum_{k=1}^nx_k=1

.
Prove or disprove that these conditions imply that

\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.

A4

1

solutions to DE system are periodic

Assume that the system of differential equations

y'=-z^3

,

z'=y^3

with the initial conditions

y(0)=1

,

z(0)=0

has a unique solution

y=f(x)

,

z=g(x)

defined for real

x

. Prove that there exists a positive constant

L

such that for all real

x

,

f(x+L)=f(x),\enspace g(x+L)=g(x).

A3

1

arctan integral

Evaluate

\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx.

A2

1

converge of infinite faulhaber sum

For positive real

x

, let

B_n(x)=1^x+2^x+\ldots+n^x.

Prove or disprove the convergence of

\sum_{n=2}^\infty\frac{B_n(\log_n2)}{(n\log_2n)^2}.

A1

1

centroid of region of R2

Let

V

be the region in the Cartesian plane consisting of all points

(x,y)

satisfying the simultaneous conditions

|x|\le y\le|x|+3\text{ and }y\le4.

Find the centroid of

V

.

B6

1

the area of triangles

Denote by

S(a,b,c)

the area of a triangle whose lengthes of three sides are

a,b,c

Prove that for any positive real numbers

a_{1},b_{1},c_{1}

and

a_{2},b_{2},c_{2}

which can serve as the lengthes of three sides of two triangles respectively ,we have

\sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}

A5

1

about four integers

a, b, c, d

are positive integers, and

r=1-\frac{a}{b}-\frac{c}{d}

.
And,

a+c \le 1982, r \ge 0

. Prove that

r>\frac{1}{1983^3}

.

1982 Putnam

Subcontests

log sequence converges, limit is continuous

n_1n_2... divides (n_1+k)(n_2+k)...

probability c+d square, find limit

double integral with exp is polynomial

triangulating triangle and reassembling them

permutation of N, limit is 1

solutions to DE system are periodic

arctan integral

converge of infinite faulhaber sum

centroid of region of R2

the area of triangles

about four integers