# A — Calc

Just directly print the given sum.

Time Complexity: $$$O(1)$$$. Click here for my submission.

# B — Minor Change

The answer is simply the number of positions $$$i$$$ for which $$$S[i] \neq T[i]$$$. This is because we must make a move fixing each of those position, and each move can only make $$$S[i] = T[i]$$$ for one new position. Thus, we can iterate through the positions and count those where $$$S$$$ and $$$T$$$ differ, printing the total as our answer.

Time Complexity: $$$O(|S|)$$$. Click here for my submission.

# C — Tsundoku

We apply two pointers. Iterate over the number of books to be read from desk A in decreasing order, starting with the greatest number of those books we could read without exceeding $$$K$$$ minutes. Maintain the sum of all books to be read from desk A. Then, while we can do so without exceeding $$$K$$$ minutes, add the next book from desk B to the list of books to be read. We can then consider the total number of books read between desk A and desk B as a possible answer, decrement the number of books from desk A, and continue.

The basic intuition is that as we decrease the number of books we read from desk A, the number of books we read from desk B monotonically increases. By processing the number of books to be read from desk A in order, we avoid having to recompute any sums, achieving a time complexity of $$$O(N+M)$$$.

Time Complexity: $$$O(N+M)$$$. Click here for my submission.

# D — Sum of Divisors

Let's reframe the problem by considering the total contribution each divisor makes to the answer. For any given $$$i$$$, $$$i$$$ contributes $$$K$$$ to the sum for each $$$K$$$ from $$$1$$$ to $$$N$$$ such that $$$K$$$ is a multiple of $$$i$$$.

Notice that the list of these $$$K$$$ is of the form $$$i$$$, $$$2i$$$, $$$3i$$$, ..., $$$\lfloor \frac{N}{i} \rfloor i$$$, noting that the last term is the greatest multiple of $$$i$$$ less than $$$N$$$. For simplicity, let $$$M = \lfloor \frac{N}{i} \rfloor.$$$ Then, the sum of this sequence is equal to $$$i + 2i + 3i + \cdots + Mi = i (1 + 2 + \cdots + M) = \frac{iM(M+1)}{2}.$$$

Thus, by computing $$$M$$$ and the above product, we can compute each divisor's contribution to the answer in $$$O(1)$$$, giving us an $$$O(N)$$$ solution.

Time Complexity: $$$O(N)$$$. Click here for my submission.

# E — NEQ

First, let's count the number of ways to choose $$$A$$$, irrespective of $$$B$$$. There are $$$\dbinom{M}{N}$$$ ways to choose the $$$N$$$ integers to appear in $$$A$$$ and $$$N!$$$ ways to arrange them, so we have $$$\dbinom{M}{N} N!$$$ choices for $$$A$$$ in total.

Now, without loss of generality, assume the sequence $$$A$$$ is simply $$$1, 2, \cdots, N$$$, since the number of choices for $$$B$$$ doesn't depend on our sequence $$$A$$$. We want to count the number of sequences $$$B$$$ where there is no position $$$i$$$ such that $$$B_i = i$$$.

We do this using the principle of inclusion and exclusion. By PIE, we know that the number of sequences $$$B$$$ where $$$B_i \neq i$$$ for all $$$i$$$ is equal to the sum over all valid $$$j$$$ of $$$(-1)^j$$$ times the number of ways to create a sequence $$$B$$$ where at least $$$j$$$ positions $$$i$$$ have $$$B_i = i$$$.

Thus, it remains to compute the number of ways to compute the number of sequences $$$B$$$ with at least $$$j$$$ equal positions. There are $$$\dbinom{N}{j}$$$ ways to choose $$$j$$$ positions where $$$B_i = i$$$. Then, it doesn't matter what we put in the remaining positions, so there are $$$\dbinom{M-j}{N-j}$$$ ways to choose the numbers and $$$(N-j)!$$$ ways to arrange them. Thus, the total number of ways to ensure that at least $$$j$$$ positions have $$$B_i = i$$$ is $$$\dbinom{N}{j} \dbinom{M-j}{N-j} (N-j)!$$$.

By precomputing all relevant factorials and their modular inverses in $$$O(M)$$$, we can compute the above product in $$$O(1)$$$ for each $$$j$$$. We then sum these up using our PIE formula, giving us an $$$O(N+M)$$$ solution. Since $$$N \leq M$$$, this is simply $$$O(M)$$$.

Time Complexity: $$$O(M)$$$. Click here for my submission.

# F — Unfair Nim

It is well known that the second player wins a game of Nim if and only if the bitwise XOR of the numbers of stones in each pile is equal to $$$0$$$. Thus, we want to make the equation $$$A_1 \wedge A_2 \wedge A_3 \wedge \cdots \wedge A_n = 0$$$. By taking the XOR of this equation with $$$A_3 \wedge A_4 \wedge \cdots \wedge A_n$$$, we have $$$A_1 \wedge A_2 = A_3 \wedge A_4 \wedge \cdots \wedge A_n.$$$ We can immediately compute the right-hand side, since it is fixed before we make our moves. Now, we want to find the largest $$$x$$$ between $$$1$$$ and $$$A_1$$$, inclusive, such that $$$x \wedge (A_1 + A_2 - x) = A_3 \wedge A_4 \wedge \cdots \wedge A_n.$$$

We build $$$x$$$ via digit DP, determining the bits of $$$x$$$ from the most significant bit to the least significant bit. Let $$$a$$$ be the maximum possible value of $$$x$$$ thus far such that we have found a bit in which $$$x$$$ contains a $$$0$$$ while $$$A_1$$$ contains a $$$1$$$ (so, we know that no matter what bits we add to $$$a$$$ in the future, it will be less than $$$A_1$$$), or $$$-\infty$$$ if no such value exists thus far. Let $$$b$$$ be the value of $$$x$$$ such that thus far, all bits in $$$b$$$ are the same as those in $$$A_1$$$, or $$$-\infty$$$ if creating this value is impossible. Note that we cannot add a $$$1$$$ to $$$b$$$ if the corresponding bit in $$$A_1$$$ is a $$$0$$$, as this would make $$$b$$$ too large.

Let $$$S = A_3 \wedge A_4 \wedge \cdots \wedge A_n.$$$ If $$$S$$$ contains a $$$0$$$ in any given bit, we know that $$$x$$$ and $$$A_1 + A_2 - x$$$ must either both contain a bit or neither must contain a bit. We can tell which case applies by determining whether after subtracting out all bits we've assigned so far, $$$A_1 + A_2$$$ is greater than $$$2^{k+1}$$$, where $$$k$$$ is the index of the current bit. This is because the sum of all future bits will be at most $$$2^{k+1} - 2$$$, so $$$A_1 + A_2 \geq 2^{k+1}$$$ both enables us to add two $$$1$$$s at position $$$b$$$ and forces us to do so. If we add a bit to both numbers, we add $$$2^k$$$ to $$$a$$$ and $$$b$$$, but set $$$b$$$ to $$$-\infty$$$ if $$$A_1$$$ contains a $$$0$$$ in position $$$k$$$. Otherwise, we don't add anything to either $$$a$$$ or $$$b$$$, but if $$$A_1$$$ contains a $$$1$$$ in position $$$k$$$, we can set $$$a$$$ to $$$\max(a, b)$$$, since now we've found a position in which $$$x < A_1$$$.

If $$$S$$$ contains a $$$1$$$ in any given bit, we know that exactly one of $$$x$$$ and $$$A_1 + A_2 - x$$$ must contain a bit. Thus, we can add $$$2^k$$$ to $$$a$$$, because this will not make $$$a$$$ exceed $$$A_1$$$. If $$$A_1$$$ contains a $$$1$$$ in position $$$k$$$, we can also add $$$2^k$$$ to $$$b$$$, or we can alternatively assign a $$$0$$$ in order to guarantee that $$$x < A[1]$$$, allowing us to set $$$a$$$ to $$$\max(a, b)$$$.

At the end of this process, $$$a$$$ and $$$b$$$ are our possible values for $$$x$$$. Of course, if $$$a$$$ and $$$b$$$ are zero or $$$-\infty$$$, they can't work (the zero case fails because we cannot move every stone from the first pile). Moreover, we also must confirm that $$$a \wedge (A_1 + A_2 - a) = S$$$ in order to use $$$a$$$, and we must check the same equation for $$$b$$$. This is because the total bits we assign throughout our process may not equal $$$A_1 + A_2$$$, since the number of bits we assign is largely determined by $$$S$$$. (One countercase that demonstrates this issue is $$$N=3, A = 7, 5, 6$$$: our solution assigns one bit in position $$$2$$$, one bit in position $$$1$$$, and two bits in position $$$0$$$, but the total of these bits is $$$4 + 2 + 2 = 8 < 7+5.$$$) Then, we take the largest of these possible values for $$$x$$$ and subtract it from $$$A_1$$$ to get our answer.

Time Complexity: $$$O(N + \log A_i).$$$ Click here for my submission.