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, 2018

Geometry Difficulty 5.6 AIME, harder Prove it Saudi Arabia

The $n \times n \times n$ cubic is constructed from $n^{3}$ unit cubic $1 \times 1 \times 1$ such that at least one of unit cubic is black. Show that we can always cut the $n \times n \times n$ cubic into rectangular boxes such that each box contains exactly one black unit cubic.

Solution

We shall prove the problem also true for all brick $a \times b \times c$ for any positive integers $a, b, c$ by induction on the number of black boxes.

First, if the brick contains only one black cubic then no any cut is needed. Suppose that it contains at least two black cubics.

We choose a plane that divide the brick into two smaller bricks such that each smaller brick contains exactly one black cubic.

By induction hypothesis, we can divide each smaller bricks to make all black cubics are separated, then we are done!

Remark. This problem is easy as long as we notice that the problem is not about "big cubic", it's true for any arbitrary brick.

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