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Print51st Ukrainian National Mathematical Olympiad, 4th Round
Ukraine counting and probability
Problem
An equilateral triangle with the side length is divided into small equilateral triangles with the side length , as it is shown on fig. 28. Parallelograms with side lengths and are cut from the triangle along the grid lines. What is the greatest number of parallelograms that can be cut in this way?
Fig. 28
Solution
Paint the small triangles in black and white as it is shown on fig. 29. Since every parallelogram with the side lengths and contains two white triangles, and the total number of white triangles is , the number of parallelograms does not exceed .
On the fig. 30 it is shown how one can cut parallelograms.
On the fig. 30 it is shown how one can cut parallelograms.
Final answer
10
Techniques
Coloring schemes, extremal argumentsInvariants / monovariantsCounting two ways