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Print50th Mathematical Olympiad in Ukraine, Third Round (January 23, 2010)
Ukraine 2010 geometry
Problem
Cube with side length is partitioned into unit cubes. The external slice of cubes is colored with yellow, the next slice (all cubes that share a vertex with the external slice) is colored with blue. Slice of cubes that touches the previous slice is colored with yellow and so on. Find the number of yellow and blue cubes.
Solution
Obviously, there are slices of cubes. It is easy to see that the external yellow slice has height , the next blue - , the next yellow - , the next blue - , the next yellow - and the last one (6th slice) blue has height , e.g. it's a unit cube. Further, we can calculate the number of cubes in each slice in such a way:
5th slice, yellow, with height , has unit cubes.
4th slice, blue, with height , has unit cubes.
3rd slice, yellow, with height , has unit cubes.
2nd slice, blue, with height , has unit cubes.
1st slice, yellow, with height , has unit cubes.
Finally, the number of blue cubes is , and the number of yellow cubes is .
5th slice, yellow, with height , has unit cubes.
4th slice, blue, with height , has unit cubes.
3rd slice, yellow, with height , has unit cubes.
2nd slice, blue, with height , has unit cubes.
1st slice, yellow, with height , has unit cubes.
Finally, the number of blue cubes is , and the number of yellow cubes is .
Final answer
yellow: 846, blue: 485
Techniques
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