Numbers in a 2012 x 2012 board
Source: 2012 Baltic Way, Problem 7
November 22, 2012
modular arithmeticanalytic geometrynumber theory unsolvednumber theory
Problem Statement
On a board, some cells on the top-right to bottom-left diagonal are marked. None of the marked cells is in a corner. Integers are written in each cell of this board in the following way. All the numbers in the cells along the upper and the left sides of the board are 1's. All the numbers in the marked cells are 0's. Each of the other cells contains a number that is equal to the sum of its upper neighbour and its left neighbour. Prove that the number in the bottom right corner is not divisible by 2011.