3

Part of 1999 Tournament Of Towns

Problems(5)

the same numbers are written on the first and the third boards

Source: Tournament Of Towns Spring 1999 Junior 0 Level p3

Several positive integers

a_0 , a_1 , a_2 , ... , a_n

are written on a board. On a second board, we write the amount

b_0

of numbers written on the first board, the amount

b_1

of numbers on the first board exceeding

1

b_2

of numbers greater than

2

, and so on as long as the

b

s are still positive. Then we stop, so that we do not write any zeros. On a third board we write the numbers

c_0 , c_1 , c_2 , ...

. using the same rules as before, but applied to the numbers

b_0 , b_1 , b_2 , ...

of the second board. Prove that the same numbers are written on the first and the third boards.
(H. Lebesgue - A Kanel)

combinatoricsSum

TOT 1999 Spring AJ3 game with 1999 digits by only 0 and 1

Source:

Two players play the following game. The first player starts by writing either

0

1

and then, on his every move, chooses either

0

1

and writes it to the right of the existing digits until there are

1999

digits. Each time the first player puts down a digit (except the first one) , the second player chooses two digits among those already written and swaps them. Can the second player guarantee that after his last move the line of digits will be symmetrical about the middle digit?
(I Izmestiev)

combinatoricsgamegame strategy

x^3 + y and x + y^3 are divisible by x^2 + y^2

Source:

(x, y)

of integers satisfying the following condition: each of the numbers

x^3 + y

x + y^3

is divisible by

x^2 + y^2

number theorydividesdivisibleSum of Squares

TOT 1999 Autumn OJ13 n lines intersect exactly 1999

Source:

n

straight lines in the plane such that each intersects exactly

1999

of the others . Find all posssible values of

n

.
(R Zhenodarov)

linescombinatorial geometrycombinatorics

TOT 1999 Autumn AS5 1, 2 , ... , n divided into 2 groups

Source:

(a) The numbers

1, 2,... , 100

are divided into two groups so that the sum of all numbers in one group is equal to that in the other. Prove that one can remove two numbers from each group so that the sums of all numbers in each group are still the same. (b) The numbers

1, 2 , ... , n

are divided into two groups so that the sum of all numbers in one group is equal to that in the other . Is it true that for every such

n > 4

one can remove two numbers from each group so that the sums of all numbers in each group are still the same?
(A Shapovalov) [(a) for Juniors, (a)+(b) for Seniors]

partitionSumcombinatoricsSubset