As shown below, I've cut a large, $5 \times 9$ rectangle into 10 smaller rectangles such that every one of them has integer side lengths.

Is it possible that the 10 small rectangles have all distinct areas?

Yes, it's possible
No, it's not possible

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Consider ten rectangles, with distinct (integer) areas, such that the sum of the areas is the smallest. It is clear that it becomes the smallest, when the areas of the rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ . The sum of the areas is equal to $1+2+3+4+5+6+7+8+9+10=55>45.$

Therefore it is not possible.