5

Part of 2000 Tournament Of Towns

Problems(4)

TOT 2000 Spring AJ5 knights on a 5 x 5 chessboard

Source:

What is the largest number of knights that can be put on a

5 \times 5

chess board so that each knight attacks exactly two other knights?
(M Gorelov)

combinatoricsChessboard

TOT 2000 Spring AS5 N consecutive, sum of digits of k-th divisible by k

Source:

What is the largest number

N

for which there exist

N

consecutive positive integers such that the sum of the digits in the

k

-th integer is divisible by

k

1 \le k \le N

consecutivenumber theorydividesdivisible

TOT 2000 Autumn AJ5 weight of 11111 grams

Source:

11111

grams is placed in the left pan of a balance. Weights are added one at a time, the first weighing

1

gram, and each subsequent one weighing twice as much as the preceding one. Each weight may be added to either pan. After a while, equilibrium is achieved. Is the

16

gram weight placed in the left pan or the right pan?
( AV Kalinin)

weighingscombinatorics

TOT 2000 Autumn AS5 bw on a m x n table

Source:

Each of the cells of an

m \times n

table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones.
(A Shapovalov)

combinatoricsrectangle tabletable