Olympiad problems

2012 Tournament of Towns, 2

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

1957 Moscow Mathematical Olympiad, 349

Tags: rectangle table , table , sum , combinatorics , geometry

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

2000 Tournament Of Towns, 5

Tags: combinatorics , rectangle table , table

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)