Third Millennium International Mathematical Olympiad 2009
Grade 10-12
Problem 4

By definition, the shortest pass between two points on the surface of a cube is the shortest broken line on the surface of the cube joining the two points. (Which is a straight line segment whenever the given points lie on the same face.) Three shortest passes joining three points split the surface of a cube into two regions. The region with the smallest area is called a triangle. Find the largest area of a triangle on the surface of a unit cube.

Solution by Carles Boix, Princeton, NJ, USA.

The maximum area will be achieved when both regions are equal. Therefore, the maximum area will be half of the surface area of the unit cube. Half of the surface area is 3 units².

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Third Millennium International Mathematical Olympiad 2009 Grade 10-12 Problem 4

Third Millennium International Mathematical Olympiad 2009
Grade 10-12
Problem 4