A - Always Increasing

Hint 1

First, fix $i$ and consider "when does $A_i < A _ {i + 1}$ always hold?"

Hint 2

The condition can be rephrased as: for some $c_i$, "$A_i+c_i < A_{i+1}$ holds in the initial state." If such $c_i$ can be found, the sum is minimized by defining the initial state so that $A_1=1$ and $A_{i+1}=A_i+c_i+1$ holds.

Editorial: https://atcoder.jp/contests/arc210/editorial/14586

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B - Remove Median Operations

Hint 1

Consider which elements remain without being deleted. For example, the smallest among $A_1, A_2, \ldots, A_N, B_1, B_2, \ldots, B_M$ is never deleted.

Hint 2

What remains without being deleted until the end are the smallest $N/2$ elements and the largest $N/2$ elements. The rest is a basic data structure problem.

Editorial: https://atcoder.jp/contests/arc210/editorial/14587

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C - Fair Coin Partition

Hint 1

The overall approach is to find the answer from the higher digits.

Hint 2

Suppose the answer is $10X+Y$ ($0\leq Y\leq 9$), and consider finding $X$. Then, after exchanging coin $0$ into coin $1$ as much as possible, we should solve the problem for coins $1,2,\ldots,N-1$.

Editorial: https://atcoder.jp/contests/arc210/editorial/14588

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D - Independent Set Game

Hint 1

Look for patterns where Bob can win with small graphs. The case where $N$ is odd is simpler.

Hint 2

When $N$ is even, if it contains two paths of three vertices, then Bob wins. Also, if it contains a cycle of four vertices, then Bob wins.

Hint 3

What kind of graph contains at most one path of three vertices and does not contain a cycle of four vertices? In the editorial, we investigate by case analysis on the number of vertices with degree $3$ or more.

Editorial: https://atcoder.jp/contests/arc210/editorial/14589

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E - Subset Sum Gaps

Hint 1

There is a simple solution of finding the sorted sequence of subset sums of $(A_1,\ldots,A_i)$ in order $i=1,2,3,\ldots$. However, if we do this as is, there are too many types of subset sums.

Hint 2

Among the subset sums, we don't need to maintain all values accurately. For elements whose exact values are unnecessary, change the values appropriately to reduce the number of types of subset sums to be maintained.

Editorial: https://atcoder.jp/contests/arc210/editorial/14590

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F - ± ab

Hint 1

It is good to first take one $x_i,y_i,z_j,w_j$ such that $ax_i+by_i=A_i$, $az_j+bw_j=B_j$.

Hint 2

By appropriate swapping, assume $bs\geq at$. Then, we can assume that $\pm a$ operations are performed at most $b$ times.

Hint 3

By reducing it to queries of the type "find the sum of values in a rectangular region," we can speed up the computation.

Editorial: https://atcoder.jp/contests/arc210/editorial/14591