A - Swap Digit

Hint 1

Consider the case \(N=2\).

Hint 2

Decompose \(A \times B\) into some number of \((10^ix+y)(10^iz+w)\).

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5592

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B - New Place

Hint 1

A necessary condition for us to be able to make \(S\) and \(T\) equal is: you get the same string from sorting \(S\) and \(T\).

Hint 2

If we can make \(S\) and \(T\) equal, there is a way to do it in \(N-1\) operations. Why?

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5593

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C - Roller

Hint 1

Consider the case \(A=(1,2,\dots,N)\).

Hint 2

After one operation, \(A\) has at most \(N-1\) different integers.

Hint 3

After one operation, you can freely rotate \(A\).

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5597

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D - A + B > C ?

Hint 1

Find \(i\) such that \(P_i = 1\).

Hint 2

Use a sorting algorithm.

Hint 3

For different integers \(a\) and \(b\), we have \(a+1 > b\) if and only if \(a > b\).

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5598

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E - Reverse and Inversion

Hint 1

We have \(f(P) = \sum_{i=1}^{N} i(i-P_i)\).

Hint 2

For an element that gets involved in one or more operations, there is a certain fact.

Hint 3

The above fact is: the distribution of its position will be symmetrical.

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5599

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F - Dice Game

Hint 1

Find the EGF of the probability generating function for the probability that all sides have shown up in \(i\) rolls.

Hint 2

Convert the computed EGF into the OGF.

Hint 3

For a probability generating function \(f(x)\), examine \([x^k]f(e^x)\).

Editorial \(\rightarrow\) https://atcoder.jp/contests/arc154/editorial/5602