Competition problem — Combinatorics

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Combinatorics Difficulty 5.5 AIME, harder Find the answer

The constructor consists of tiles measuring $1 \times 3$ and $1 \times 4$ . Using all the available tiles, Fedya built two rectangles with dimensions $2 \times 6$ and $7 \times 8$ . His brother Anton took one tile from each of the built rectangles. Will Fedya be able to assemble a rectangle of size 12 × 5 from the remaining tiles?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Suppose Fedia managed to assemble the required rectangle, then after Anton's actions, there should have been $12 \times 5=60$ cells left. The total area of all the tiles is $2 \times 6+7 \times 8=68$ cells. Note that from the rectangle $2 \times 6$ , only a tile of size $1 \times 3$ can be taken. Depending on which tile Anton took from the rectangle $7 \times 8$ , either 62 cells or 61 cells could remain. This is more than 60, but if at least one more tile is removed, then less than 60 will remain. Therefore, Fedia will not be able to assemble the specified rectangle.

## Answer

He will not be able to.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.