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## 1734A — Select Three Sticks

We first sort the array $$$a$$$ in non-decreasing order.

Denote the indices of the elements that we choose from $$$a$$$ to be $$$x$$$, $$$y$$$, and $$$z$$$, where $$$1 \le x < y < z \le n$$$, and the final value (after performing the operations) of the concerned elements to be $$$v$$$.

The minimum required number of operations is then $$$|a_x-v|+|a_y-v|+|a_z-v|$$$. It is well-known that such expression attains its minimum value when $$$v$$$ is the median of $$$a_x$$$, $$$a_y$$$, and $$$a_z$$$. Since the array $$$a$$$ has already been sorted, it is best to assign $$$v$$$ to be $$$a_y$$$.

Our expression then becomes $$$|a_x-a_y|+|a_y-a_y|+|a_z-a_y|=(a_y-a_x)+0+(a_z-a_y)=a_z-a_x$$$. We would like to minimize the value of $$$a_z$$$, which implies $$$z$$$ should be as small as possible since $$$a$$$ is sorted. It is clear that taking $$$z=y+1$$$ would minimize the value of the expression. Similarly, we can show that we can take $$$x=y-1$$$ to minimize the value of the expression.

Therefore, the only possible values of the triplets $$$(x,y,z)$$$ are of the form $$$(t,t+1,t+2)$$$ for positive integers $$$1 \le t \le n-2$$$, and we can iterate through all such triplets and find the best one.

The time complexity is $$$\Theta(n \log n)$$$ per case due to sorting.

**Code in C++**

## 1734B — Bright, Nice, Brilliant

Note that the brightnesses of the rooms on the $$$i$$$-th floor is at most $$$i$$$. This is because in room $$$(i,1)$$$, only $$$i$$$ rooms, namely, $$$(1,1)$$$, $$$(2,1)$$$, $$$\ldots$$$, $$$(i,1)$$$ can reach to $$$(i,1)$$$ through some number of staircases.

It is also possible to find a configuration of torches in the pyramid such that the brightnesses of the rooms on the $$$i$$$-th floor is exactly $$$i$$$, i.e. it attains the upper bound.

The configuration is as follows: Room $$$(i,j)$$$ contains a torch if and only if it is the leftmost room ($$$i=1$$$) or the rightmost room ($$$i=j$$$) on the $$$i$$$-th floor.

This is valid because for all rooms $$$(i,j)$$$, it can be reached from $$$(1,1)$$$, $$$(2,1)$$$, $$$(3,1)$$$, $$$\ldots$$$, $$$(i-j+1,1)$$$ and $$$(2,2)$$$, $$$(3,3)$$$, $$$\ldots$$$, $$$(j,j)$$$. In other words, room $$$(i,j)$$$ has brightness $$$(i-j+1)+j-1=i$$$, so the pyramid is nice.

**Code in C++**

## 1734C — Removing Smallest Multiples

One operation should be used to remove every element not belonging to $$$T$$$.

Let $$$v$$$ be an element not belonging to $$$T$$$. Suppose a $$$x$$$-cost operation removes value $$$v$$$, then $$$v$$$ must be divisible by $$$x$$$. Furthermore, the multiples $$$x,2x,\cdots (k-1)x$$$ must have been already removed from $$$S$$$, where we write $$$v = kx$$$.

Since removed elements stay removed, the above is only possible if all of $$$x,2x,\cdots (k-1)x$$$ does not belong to $$$T$$$.

For each $$$v$$$, let $$$f(v)$$$ be the smallest integer $$$x$$$ satisfying the above condition. As we can always remove $$$v$$$ using a $$$v$$$-cost operation, $$$f(v) \leq v$$$ and in particular $$$f(v)$$$ exists.

The total cost must be at least $$$\sum_{i \not \in T} f(i)$$$. We claim that this cost can be achieved.

To do so, we should remove the required elements in ascending order. When removing $$$v$$$, we assume all $$$w \not\in T$$$ with $$$w<v$$$ have already been removed. At this state, an $$$f(v)$$$-cost operation would be able to remove $$$v$$$.

It remains to find the values $$$f(v)$$$. To do so efficiently, we can perform the above process in a bottom-up manner similar to the Sieve of Eratosthenes. Please refer to the code below for implementation details. The overall complexity is $$$n (1 +\frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}) = \Theta(n \log n)$$$.

**Code in C++**

## 1734D — Slime Escape

Let's call a group of slime **good** if their total health is at least $$$0$$$, or if defeating this group allows you to reach the exits.

We partition the slimes into good groups in a two-pointer like manner. To form the groups to the right, start from position $$$k$$$, then find the smallest position $$$r$$$ such that slimes from $$$k+1$$$ through $$$r$$$ form a good group. We do the same starting from $$$r+1$$$ again. Repeat this process until slimes to the right are partitioned into groups, which can be done by maintaining the sum of health. We partition the left slimes into groups in a similar way.

We can observe that in an optimal strategy, we may assume the player absorbs group-by-group.

**Proof**

For any good group, since the total health is positive, there is no drawback to absorbing a good group. In other words, whenever it is possible to absorb a good group, we will absorb it.

For each group $$$G$$$, we calculate the ``requirement'' of the group — the lowest health we can begin with, such that we can absorb the group while maintaining non-negative health at all times. The requirement of a group of slime with health $$$a_1,a_2 \cdots a_n$$$ can be expressed as

Finally, we can simply simulate the process. We repeatedly attempt to absorb good groups to the left or to the right. We keep track of the current health, initially equal to $$$a_k$$$. Whenever we consider whether to absorb a group or not, we absorb it if and only if the current health is at least as high as its requirement. Otherwise, we ignore it for now and attempt to do so for the group on the other side. If it is possible to reach a state where either all left groups or all right groups are absorbed, then we can win the game. If at some point, it is not possible to absorb the left group nor the right group, then we lose.

The overall complexity is $$$\Theta(n)$$$.

It is also possible to use a range $$$\max$$$/$$$\min$$$ segment tree form the groups instead of using two-pointers, in which case its complexity would be $$$\Theta(n \log n)$$$.

**Code in C++**

## 1734E — Rectangular Congruences

We say a matrix to be good if it satisfies the congruence condition (the second condition).

When we have a good matrix, we can add any value $$$c$$$ to a whole row while maintaining the congruence relation. The same is true for adding the same value to a whole column.

Suppose we have any good matrix $$$A$$$, then by adding $$$b_i - a_{i,i}$$$ to the $$$i$$$-th row for each of $$$i=1,2,\cdots,n$$$, we obtain a good matrix that has the desired values on the diagonal.

In fact, there are a lot of possible constructions. We present a few of them here:

$$$a_{i,j} = i \times j \pmod n$$$

$$$a_{i,j} = (i + j)^2 \pmod n$$$. This needs special handling when $$$n=2$$$.

$$$a_{i,j} = \frac{(i+j)(i+j+1)}{2} \pmod n$$$.

The coolest part is that all quadratic polynomials in the form $$$ai^2 + bij + cj^2 + di + ej + f$$$ are valid for all integers $$$a,b,c,d,e,f$$$ and $$$b \not\equiv 0 \pmod n$$$

As a bonus, we prove that the general quadratic polynomial gives a good construction.

**Proof**

Here are some extra observations that may enable one to find a good matrix more quickly.

**Some extra observations**

**Code in C++**

## 1734F — Zeros and Ones

Observe that the $$$i$$$-th character is `1' if and only if $$$i$$$ has an odd number of set bits in its binary representation. Both solutions make use of this fact.

The constraints allows solutions of up to $$$\Theta(q \log^3 n)$$$. Yet, both of the model solution runs in $$$\Theta(\log n)$$$.

### 1. Digit DP solution

The question can be reformulated as follows: How many integers $$$x$$$ between $$$0$$$ and $$$m-1$$$ inclusive have the property that the total number of set bits of $$$x$$$ and $$$x+n$$$ is an odd number?

This can be solved with digit DP. We process the bit position from $$$\lceil \log(\max) \rceil$$$ down to $$$0$$$. We maintain three states:

$$$\text{ans}$$$, a boolean value;

$$$\text{trailzeros}$$$, an integer between $$$0$$$ and $$$\lceil \log(\max) \rceil$$$ inclusive; and

$$$\text{under}$$$, a boolean value.

We can thus conclude the following: the number of trailing zeros is all we need to decide the answer.

After processing each bit $$$k$$$, we should have the following: the number of integers $$$x$$$ between $$$0$$$ and $$$\lfloor \frac{m}{2^k} \rfloor$$$ inclusive which have the following property:

the total number of set bits of $$$x$$$ and $$$x + \lfloor \frac{n}{2^k} \rfloor$$$ is equal to $$$\text{ans} \mod 2$$$;

the number of trailing `1's of $$$x + \lfloor \frac{n}{2^k} \rfloor$$$ is equal to $$$\text{trailzeros}$$$;

the boolean value $$$[x < \lfloor \frac{m}{2^k} \rfloor]$$$ (where $$$[]$$$ is the Iverson bracket).

Now onto the transitions. Suppose we are adding the $$$(k-1)$$$-th digit, and let $$$d$$$ be the new digit of $$$x$$$, and $$$z$$$ be the $$$(k-1)$$$-th digit of $$$n$$$.

If $$$z+d = 0$$$, then $$$(\text{ans},\text{trailzeros})$$$ after digit $$$k$$$ will be transited to $$$(\text{ans},0)$$$ after digit $$$k-1$$$;

if $$$z+d = 1$$$, then $$$(\text{ans},\text{trailzeros})$$$ after digit $$$k$$$ will be transited to $$$((\text{ans} + z +d) \mod 2, \text{trailzeros}+1)$$$ after digit $$$k-1$$$;

if $$$z+d =2$$$, then $$$(\text{ans},\text{trailzeros})$$$ after digit $$$k$$$ will be transited to $$$((\text{ans} + z + \text{trail} + 1) \mod 2, 0)$$$ after digit $$$k-1$$$

The final answer is the total number of values for which $$$\text{ans} =1 $$$ and $$$\text{under} = 1$$$.

The above solution runs in $$$\Theta(\log^2 (\max))$$$ per query. There is a simple way to optimize this to $$$\Theta(\log(\max))$$$: note that we only need to keep parity of $$$\text{trailzero}$$$.

There are many other digit DP approaches that give similar time complexity. The constraints should allow most of them pass.

**Code in Kotlin**

### 2. Recursive Solution, by darkkcyan

Define the function $$$f(x) := $$$ the parity of bit one in the number $$$x$$$.

We have thus reworded the statement into evaluating the follow expression:

The formula can be further transformed as:

since $$$\left[ f(a) \ne f(b) \right] = f(a \oplus b)$$$ holds true for all non-negative integers $$$a$$$ and $$$b$$$.

Imagine we construct a grid and assign the value at row $$$r$$$ and column $$$c$$$ to be $$$f(r \oplus c)$$$. Then, $$$T$$$ is sum of a diagonal of length $$$k$$$ which starts at either $$$(0, n)$$$ or $$$(n, 0)$$$. Without loss of generality, we use $$$(0, n)$$$ in this editorial.

The grid can be constructed similarly to the way we construct the string $$$S$$$. We start with a $$$1$$$-by-$$$1$$$ matrix $$$M_0=\begin{bmatrix} 0 \end{bmatrix}$$$.

Then, the matrix $$$M_i$$$ of size $$$2^i \times 2^i$$$ can be constructed as follows:

where $$$\overline{M_{i - 1}}$$$ is the matrix $$$M_{i - 1}$$$ but with flipped bits.

Here is another way of constructing the grid: let $$$C_i$$$ be an infinite chess board with alternating colors, similar to a regular chess board, but with each of the cells being size $$$2^i \times 2^i$$$. For example, $$$C_0$$$, $$$C_1$$$ and $$$C_2$$$ in an $$$8$$$-by-$$$8$$$ grid is as follows:

**Illustration**

We claim that our grid is the $$$\text{xor}$$$ of all chess board $$$C_i$$$. The proof is easy: $$$C_i$$$ is constructed by $$$\text{xor}$$$-ing the $$$i$$$-th bit of the the row and column number.

We are therefore motivated to proceed in the following way: if we drop the least significant bit (by making it to be $$$0$$$), we are still solving a very similar problem to the original problem, because dropping the first bit is similar to removing $$$C_0$$$. And when we shift $$$C_i$$$ to $$$C_{i - 1}$$$, it is a recursion of the same problem!

Going back to the problem, where we are computing sum of a diagonal of length $$$k$$$. If $$$k$$$ is odd, we can make it odd by adding the last element to the result and decreasing $$$k$$$ by one. Now, $$$k$$$ is even, and we can utilize the recurrence as follows:

remove $$$C_0$$$.

scale the board down by 2 (including $$$n$$$ and $$$k$$$). By doing so, $$$C_i$$$ becomes $$$C_{i - 1}$$$.

solve the new problem.

scale the board up again and add $$$C_0$$$ back.

from the result of the scaled down problem, some how calculate the result of the original problem

The result of the scaled down problem is the number of $$$2$$$-by-$$$2$$$ cells with value $$$1$$$. From the number of $$$2$$$-by-$$$2$$$ cells with value $$$1$$$, we compute the number of cells with value $$$0$$$ as well. It is not hard to observe that it crosses the $$$2$$$-by-$$$2$$$ cells at all places. The only thing that matters is the parity of $$$n$$$.

If $$$n$$$ is even, then the diagonal crosses the diagonal of the $$$2$$$-by-$$$2$$$ cells. In the scaled-down version, the diagonal is still a single diagonal starting at $$$(0, \frac{n}{2})$$$; otherwise,

if $$$n$$$ is odd, it crosses the corner of the $$$2$$$-by-$$$2$$$ cells. In the scaled-down version, the diagonal is actually $$$2$$$ neighboring diagonals starting at $$$(0, \frac{n-1}{2})$$$ and $$$(0, \frac{n+1}{2})$$$.

Also, the $$$2$$$-by-$$$2$$$ cells with values $$$0$$$ and $$$1$$$ respectively will also have the form:

**Illustration**

From here we have everything we need to compute the result of the original problem.

Overall, the number of states we have to visit is $$$\Theta(\log k)$$$.

**Code in Python**