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4 problems match “induction”. Clear

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1. Possible values of f(2018) ●●●○○ 50 ptsLet $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be an even function with $f(1)=1$ and $f(2017) \neq 1$. For all integers $x, y$, $2f(x+y)-f(x)-f(y) \leqslant |f(x)-f(y)|$. Find the value of $f(2018)$.
2. Coin flips moving two numbers together ●●●○○ 50 ptsSteph and Jeff each start with the number 4, and Travis is flipping a coin. Every time he flips heads, Steph replaces her number $x$ with $2x-1$, and Jeff replaces his number $y$ with $y+8$. Every time he flips tails, Steph replaces her number $x$ with $\frac{x+1}{2}$, and Jeff replaces his number $y$ with $y-3$. After some (positive) number of coin flips, Steph and Jeff end up with the same number, which is below 2012. How many times was the coin flipped?
3. Minimum operations to clear 2021 piles ●●●○○ 50 ptsThere are 2021 piles of stones, with the number of stones in each pile being $1, 2, \ldots, 2021$ respectively. In one operation you select any number of piles and remove the same number of stones from each selected pile. What is the minimum number of operations required to remove all the stones?
4. 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?