MathDB

1

expected value of sum equals integral

(x_1,x_2,\ldots,x_n)

be a point chosen at random in the

n

-dimensional region defined by

0<x_1<x_2<\ldots<x_n<1

, denoting

x_0=0

and

x_{n+1}=1

. Let

f

be a continuous function on

[0,1]

with

f(1)=0

. Show that the expected value of the sum

\sum_{i=0}^n(x_{i+1}-x_i)f(x_{i+1})

is

\int^1_0f(t)P(t)dt

., where

P

is a polynomial of degree

n

, independent of

f

, with

0\le P(t)\le1

for

0\le t\le1

.

B5

1

upper bound of side ratio in trapezoid

Label the vertices of a trapezoid

T

inscribed in the unit circle as

A,B,C,D

counterclockwise with

AB\parallel CD

. Let

s_1,s_2,

and

d

denote the lengths of

AB

,

CD

, and

OE

, where

E

is the intersection of the diagonals of

T

, and

O

is the center of the circle. Determine the least upper bound of

\frac{s_1-s_2}d

over all

T

for which

d\ne0

, and describe all cases, if any, in which equality is attained.

B4

1

subsets of infinite set have finite intersection

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

B3

1

f'(x)=-3f(x)+6f(2x), find integral of x^nf(x)

Let

f:[0,\infty)\to\mathbb R

be differentiable and satisfy

f'(x)=-3f(x)+6f(2x)

for

x>0

. Assume that

|f(x)|\le e^{-\sqrt x}

for

x\ge0

. For

n\in\mathbb N

, define

\mu_n=\int^\infty_0x^nf(x)dx.

a.

Express

\mu_n

in terms of

\mu_0

.

b.

Prove that the sequence

\frac{3^n\mu_n}{n!}

always converges, and the the limit is

0

only if

\mu_0

.

A6

1

power series in F2

Let

\alpha=1+a_1x+a_2x^2+\ldots

be a formal power series with coefficients in the field of two elements. Let

a_n=\begin{cases}1&\text{if every block of zeroes in the binary expansion of }n\text{ has an even number of zeroes}\\0&\text{otherwise}\end{cases}

(For example,

a_{36}=1

since

36=100100_2

) Prove that

\alpha^3+x\alpha+1=0

.

A5

1

point in 2n+1-gon, distance from two vertices

Show that we can find

\alpha>0

such that, given any point

P

inside a regular

2n+1

-gon which is inscribed in a circle radius

1

, we can find two vertices of the polygon whose distance from

P

differ by less than

\frac1n-\frac\alpha{n^3}

.

A4

1

A gambling game

Is there a gambling game with an honest coin for two players, in which the probability of one of them winning is

\frac{1}{{\pi}^e}

.

A3

1

Putnam 1989: A3 (Complex Numbers and Unit Modulus Proof)

Prove that all roots of 11z^{10} \plus{} 10iz^9 \plus{} 10iz \minus{}11 \equal{} 0 have unit modulus (or equivalent |z| \equal{} 1).

A1

1

10101...101

How many base ten integers of the form 1010101...101 are prime?

B2

1

Putnam 1989 B2

Let S be a non-empty set with an associative operation that is left and right cancellative (xy=xz implies y=z, and yx = zx implies y = z). Assume that for every a in S the set {a^n : n = 0,1,2...} is finite. Must S be a group? I haven't had much group theory at this point...

B1

1

Putnam 1989 B1

A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equall likely to be hit, find the probability that hte point hit is nearer to the center than any edge.

A2

1

Putnam 1989 A2

Evaluate

\int^{a}_{0}{\int^{b}_{0}{e^{max(b^{2}x^{2},a^{2}y^{2})}dy dx}}

1989 Putnam

Subcontests

expected value of sum equals integral

upper bound of side ratio in trapezoid

subsets of infinite set have finite intersection

f'(x)=-3f(x)+6f(2x), find integral of x^nf(x)

power series in F2

point in 2n+1-gon, distance from two vertices

A gambling game

Putnam 1989: A3 (Complex Numbers and Unit Modulus Proof)

10101...101

Putnam 1989 B2

Putnam 1989 B1

Putnam 1989 A2