### IMO Shortlist 2005 problem C3

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Avg: 7.0 Consider a rectangular board consisting of unit squares. Two of its unit squares are called adjacent if they have a common edge, and a path is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called non-intersecting if they don't share any common squares.

Each unit square of the rectangular board can be colored black or white. We speak of a coloring of the board if all its unit squares are colored.

Let be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.

Prove that .

Each unit square of the rectangular board can be colored black or white. We speak of a coloring of the board if all its unit squares are colored.

Let be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.

Prove that .

Source: Međunarodna matematička olimpijada, shortlist 2005