Maths Olympiad Prep

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Combinatorics Difficulty 5.5 AIME, harder Prove it Brazil

Carlitos has several pieces formed by four unit squares, shaped as an L:
Figure 1
He assembles bigger figures with these pieces, making them share one or more sides of the little squares. In the following example, the figure in the left was assembled by two pieces sharing a unit side. The figures are not allowed to have holes.
Figure 2
Figure 3

a) Draw a figure with perimeter 1414.

b) Describe how is it possible to obtain a figure with perimeter 20102010.

c) Is it possible to obtain a figure with odd perimeter? Justify your answer.

Solution

a) For example,
Figure 4
(of course, there are other possibilities)

b) For example,
Figure 5
which consists of a rectangle with dimensions 4×10014 \times 1001 minus a square of side 22.

c) No, it's not possible, because 20092009 is odd. Each piece has perimeter 1010, which is even, and each junction between a piece and a figure adds to the perimeter 1010 minus twice the number of the common sides of the piece and the figure. Hence the perimeter of every figure is always even.

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