MathDB

1

limit sequence of polygons, trisecting sides

(P_n),n\ge0,

is defined inductively as follows.

P_0

is an equilateral triangle with side length

1

. Once

P_n

has been determined, its sides are trisected; the vertices of

P_{n+1}

are the interior trisection points of the sides of

P_n

. Express

\lim_{n\to\infty}[P_n]

in the form

\frac{\sqrt a}b

, where

a,b

are integers.

B5

1

parity of #1s in binary expansion

For each nonnegative integer

k

, let

d(k)

denote the number of

1

's in the binary expansion of

k

. Let

m

be a positive integer. Express

\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m

in the form

(-1)^ma^{f(m)}g(m)!

, where

a

is an integer and

f

and

g

are polynomials.

B4

1

centroid of regions is same as average value of function

Find, with proof, all real-valued functions

y=g(x)

defined and continuous on

[0,\infty)

, positive on

(0,\infty)

, such that for all

x>0

the

y

-coordinate of the centroid of the region

R_x=\{(s,t)\mid0\le s\le x,\enspace0\le t\le g(s)\}

is the same as the average value of

g

on

[0,x]

.

B3

1

existence of operator on set which is nonassociative and invertible

Prove or disprove the following statement: If

F

is a finite set with two or more elements, then there exists a binary operation

*

on

F

such that for all

x,y,z

in

F

,

(\text i)

x*z=y*z

implies

x=y

(\text{ii})

x*(y*z)\ne(x*y)*z

B1

1

recursion for 1!+2!+...+n!

Let

n

be a positive integer, and define

f(n)=1!+2!+\ldots+n!

. Find polynomials

P

and

Q

such that

f(n+2)=P(n)f(n+1)+Q(n)f(n)

for all

n\ge1

.

A6

1

last nonzero digit of (5^a+5^b+...)!, find periodicity

Let

n

be a positive integer, and let

f(n)

denote the last nonzero digit in the decimal expansion of

n!

.

(\text a)

Show that if

a_1,a_2,\ldots,a_k

are distinct nonnegative integers, then

f(5^{a_1}+5^{a_2}+\ldots+5^{a_k})

depends only on the sum

a_1+a_2+\ldots+a_k

.

(\text b)

Assuming part

(\text a)

, we can define

g(s)=f(5^{a_1}+5^{a_2}+\ldots+5^{a_k}),

where

s=a_1+a_2+\ldots+a_k

. Find the least positive integer

p

for which

g(s)=g(s+p),\enspace\text{for all }s\ge1,

or show that no such

p

exists.

A4

1

maximum area of pentagon with AC||BD

A convex pentagon

P=ABCDE

is inscribed in a circle of radius

1

. Find the maximum area of

P

subject to the condition that the chords

AC

and

BD

are perpendicular.

A2

1

infinite sum involving powers

Express

\sum_{k=1}^\infty\frac{6^k}{(3^{k+1}-2^{k+1})(3^k-2^k)}

as a rational number.

A1

1

point distance of 1 from rectangular prism, find volume

Let

A

be a solid

a\times b\times c

rectangular brick, where

a,b,c>0

. Let

B

be the set of all points which are a distance of at most one from some point of

A

. Express the volume of

B

as a polynomial in

a,b,

and

c

.

A3

1

Putnam A3 1984

Let

n

be a positive integer. Let

a,b,x

be real numbers, with

a \neq b

and let

M_n

denote the

2n x 2n

matrix whose

(i,j)

entry

m_{ij}

is given by

m_{ij}=x

if

i=j

,

m_{ij}=a

if

i \not= j

and

i+j

is even,

m_{ij}=b

if

i \not= j

and

i+j

is odd. For example

M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}

. Express

\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }

as a polynomial in

a,b

and

n

.
P.S. How write in latex

m_{ij}=...

with symbol for the system (because is multiform function?)

A5

1

Putnam 1984/A5

Putnam 1984/A5) Let

R

be the region consisting of all triples

(x,y,z)

of nonnegative real numbers satisfying

x+y+z\leq 1

. Let

w=1-x-y-z

. Express the value of the triple integral

\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz

in the form

a!b!c!d!/n!

where

a,b,c,d

and

n

are positive integers. [hide="A solution"]\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz
where

Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}

, which is a Dirichlet integral giving
4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}

B2

1

Minimum value

Find the minimum value of

(u-v)^2+\left(\sqrt{2-u^2}-\frac{9}{v}\right)^2

for

0<u<\sqrt{2}

and

v>0

1984 Putnam

Subcontests

limit sequence of polygons, trisecting sides

parity of #1s in binary expansion

centroid of regions is same as average value of function

existence of operator on set which is nonassociative and invertible

recursion for 1!+2!+...+n!

last nonzero digit of (5^a+5^b+...)!, find periodicity

maximum area of pentagon with AC||BD

infinite sum involving powers

point distance of 1 from rectangular prism, find volume

Putnam A3 1984

Putnam 1984/A5

Minimum value