A - Colorful Intervals

Hint 1

Since $A_L,A_{L+1},\ldots,A_R$ must be distinct, at least $R-L+1$ distinct positive integers are needed.

Hint 2

With $d = \max_j(R_j-L_j+1)$, it is possible to satisfy the condition using $d$ distinct positive integers. Arrange the positive integers so that each one appears at most once every $d$ positions when the elements are read in index order.

Editorial: https://atcoder.jp/contests/arc222/editorial/21478

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B - Circular RPS

Hint 1

Try splitting the problem by cutting at positions where losers appear consecutively.

Hint 2

Perform casework on whether each of the patterns where A is a winner, B is a winner, and C is a winner appears at least once.

Hint 3

If you are getting Wrong Answer, writing a random test to check what kinds of cases you are getting wrong is a good approach.

Editorial: https://atcoder.jp/contests/arc222/editorial/21479

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C - 2 Directions vs 4 Directions

Hint 1

Since Bob can undo Alice's moves, Alice must ensure that Bob's destination is always a white cell.

Hint 2

Considering the cases where Bob keeps moving upward and where Bob keeps moving downward gives a necessary condition for Alice to win.

Editorial: https://atcoder.jp/contests/arc222/editorial/21480

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D - Shift and Add

Hint 1

At the point when $X_i$ has been determined, the digits below the $10^9$ place are at most $2^9$ possibilities. The upper digits are approximately fixed at this point. That is, essentially, the decimal representation shifts upward as-is.

Hint 2

However, the upper digits may increase by at most $+1$ due to a future carry from the lower digits. Perform DP by computing both the minimum digit sum of the upper digits and the minimum digit sum of the upper digits incremented by $1$.

Editorial: https://atcoder.jp/contests/arc222/editorial/21481

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E - XOR Matching

Hint 1

Think about a solution for the case where the number of distinct integers appearing in $A$ is small.

Hint 2

Think about a solution for the case where each integer appears in $A$ a small number of times (for example, when all $A_i$ are distinct).

Editorial: https://atcoder.jp/contests/arc222/editorial/21482

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F - Triple Transform

Hint 1

The difficulty of this problem lies in the fact that which of the four types of transformations is applied depends on the relative sizes of $x,y,z$. However, all four types of transformations share a common property, and discovering this property leads to the solution.

Hint 2

One common property is that all four types of transformations leave $x+y+z$ unchanged. Finding one more common property in addition to this is important.

Hint 3

Let $S=x+y+z$, and observe the values after transformation modulo $S$.

Editorial: https://atcoder.jp/contests/arc222/editorial/21484