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3 problems match “powers of two”. Clear

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1. Sizes of two sets from subset counts ●●○○○ 20 ptsLet $A$ and $B$ be two finite sets. The number of subsets of $A$ minus the number of subsets of $B$ is $112$. Find the number of elements in $A$ and in $B$.
2. Arm wrestling tournament rounds ●●●●○ 100 ptsIn an arm wrestling tournament, 510 athletes participate. A win gives 1 point and a loss gives 0 points. If the winner initially had fewer points than the opponent, the winner additionally receives one point from the loser. In each round, participants whose scores differ by at most 1 point compete. The tournament ends as soon as a sole leader is determined. What is the minimum number of rounds that need to be played?
3. Subsets with smallest element m ●●●●○ 100 ptsDetermine all values of $m$ with $1 \leq m \leq 47$ for which there are exactly $2^{15}$ subsets $X$ of the set $\{1,2,3,\ldots,47\}$ such that $m$ is the smallest element of $X$, and for every $x \in X$, either $x+m \in X$ or $x+m>47$.