MathDB

1

number of words consisting of 1998 A’s and 1998 B's

In a certain language there are only two letters,

A

and

B

. The words of this language obey the following rules: (i) The only word of length

1

is

A

; (ii) A sequence of letters

X_1X_2...X_{n+1}

, where

X_i\in \{A,B\}

for each

i

, forms a word of length

n+1

if and only if it contains at least one letter

A

and is not of the form

WA

for a word

W

of length

n

. Show that the number of words consisting of

1998 A

’s and

1998 B

’s and not beginning with

AA

equals

\binom{3995}{1997}-1

5

1

sum of integers is 1492, can sum of 7th powers be 1996,1998?

The sum of several integers (not necessarily distinct) equals

1492

. Decide whether the sum of their seventh powers can equal (a)

1996

; (b)

1998

.

4

1

100 balls, each of them touches at least 1 third of others

Is it possible to place

100

balls in space so that no two of them have a common interior point and each of them touches at least one third of the others?

2

1

community of more than 6 exchanges letters with exactly 3

In a community of more than six people each member exchanges letters with exactly three other members of the community. Show that the community can be partitioned into two nonempty groups so that each member exchanges letters with at least two members of the group he belongs to.

1

find ratio so segments are perpendicular inside equilateral

Points

K

and

L

are chosen on the sides

AB

and

AC

of an equilateral triangle

ABC

such that

BK = AL

. Segments

BL

and

CK

intersect at

P

. Determine the ratio

\frac{AK}{KB}

for which the segments

AP

and

CK

are perpendicular.

1997 Czech and Slovak Match

Subcontests

number of words consisting of 1998 A’s and 1998 B's

sum of integers is 1492, can sum of 7th powers be 1996,1998?

100 balls, each of them touches at least 1 third of others

community of more than 6 exchanges letters with exactly 3

find ratio so segments are perpendicular inside equilateral

1997 Czech and Slovak Match

Subcontests

number of words consisting of 1998 A&rsquo;s and 1998 B's

sum of integers is 1492, can sum of 7th powers be 1996,1998?

100 balls, each of them touches at least 1 third of others

community of more than 6 exchanges letters with exactly 3

find ratio so segments are perpendicular inside equilateral

number of words consisting of 1998 A’s and 1998 B's