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Part of 2020 Caucasus Mathematical Olympiad

Problems(2)

Find the number of spiders in magic transformations

Source: 2020 Caucasus Mathematical Olympiad

By one magic nut, Wicked Witch can either turn a flea into a beetle or a spider into a bug; while by one magic acorn, she can either turn a flea into a spider or a beetle into a bug. In the evening Wicked Witch had spent 20 magic nuts and 23 magic acorns. By these actions, the number of beetles increased by 5. Determine what was the change in the number of spiders. (Find all possible answers and prove that the other answers are impossible.)

Does there exist such a finite set ?

Source: Caucasus 2020 Seniors Day 1/ P1

Determine if there exists a finite set

A

of positive integers satisfying the following condition: for each

a\in{A}

at least one of two numbers

2a

\frac{a}{3}

A