MathDB

1

monochromatic Cn in 2-coloring for arbitrary n

An enormous party has an infinite number of people. Each two people either know or don't know each other. Given a positive integer

n

, prove there are

n

people in the party such that either they all know each other, or nobody knows each other (so the first possibility means that if

A

and

B

are any two of the

n

people, then

A

knows

B

, whereas the second possibility means that if

A

and

B

are any two of the

n

people, then

A

does not know

B

).

Problem 4

1

domino tiling 9x9 board without two corners and center (VTRMC 2004 P4)

A

9\times9

chess board has two squares from opposite corners and its central square removed. Is it possible to cover the remaining squares using dominoes, where each domino covers two adjacent squares? Justify your answer.

Problem 2

1

given recurrence, find a_(6n)

A sequence of integers

\{f(n)\}

for

n=0,1,2,\ldots

is defined as follows:

f(0)=0

and for

n>0

,

\begin{matrix}f(n)=&f(n-1)+3,&\text{if }n=0\text{ or }1\pmod6,\\&f(n-1)+1,&\text{if }n=2\text{ or }5\pmod6,\\&f(n-1)+2,&\text{if }n=3\text{ or }4\pmod6.\end{matrix}

Derive an explicit formula for

f(n)

when

n\equiv0\pmod6

, showing all necessary details in your derivation.

Problem 3

1

six-letter word with three letters doesn't contain A (VTRMC 2004 P3)

A computer is programmed to randomly generate a string of six symbols using only the letters

A,B,C

. What is the probability that the string will not contain three consecutive

A

's?

Problem 7

1

infinite abs sum is divergent for sequence in R+

Let

\{a_n\}

be a sequence of positive real numbers such that

\lim_{n\to\infty}a_n=0

. Prove that

\sum^\infty_{n=1}\left|1-\frac{a_{n+1}}{a_n}\right|

is divergent.

Problem 5

1

int^x_0 sin(t^2-t+x)dt=f(x), find f''(x)+f(x)

Let

f(x)=\int^x_0\sin(t^2-t+x)dt

. Compute

f''(x)+f(x)

and deduce that

f^{(12)}(0)+f^{(10)}(0)=0

.

Problem 1

1

M invertible -> N invertible for I&A\\B&C

Let

I

denote the

2\times2

identity matrix

\begin{pmatrix}1&0\\0&1\end{pmatrix}

and let

M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}

where

A,B,C

are arbitrary

2\times2

matrices which entries in

\mathbb R

, the real numbers. Thus

M

and

N

are

4\times4

matrices with entries in

\mathbb R

. Is it true that

M

is invertible (i.e. there is a

4\times4

matrix

X

such that

MX=XM=I

) implies

N

is invertible? Justify your answer.

2004 VTRMC

Subcontests

monochromatic Cn in 2-coloring for arbitrary n

domino tiling 9x9 board without two corners and center (VTRMC 2004 P4)

given recurrence, find a_(6n)

six-letter word with three letters doesn't contain A (VTRMC 2004 P3)

infinite abs sum is divergent for sequence in R+

int^x_0 sin(t^2-t+x)dt=f(x), find f''(x)+f(x)

M invertible -> N invertible for I&A\\B&C

2004 VTRMC

Subcontests

monochromatic Cn in 2-coloring for arbitrary n

domino tiling 9x9 board without two corners and center (VTRMC 2004 P4)

given recurrence, find a_(6n)

six-letter word with three letters doesn't contain A (VTRMC 2004 P3)

infinite abs sum is divergent for sequence in R+

int^x_0 sin(t^2-t+x)dt=f(x), find f''(x)+f(x)

M invertible -&gt; N invertible for I&amp;A\\B&amp;C

M invertible -> N invertible for I&A\\B&C