There are two teams, each consisting of N players, playing a round of Counter-Strike 2 (CS2).

You are given the total damage dealt by the players of one team to the opposing team. Specifically, you are given an array damage, where damage[i] is the total damage dealt by the i-th player to enemy players only.

Players cannot deal damage to their own teammates or themselves.

Each player on the round starts with exactly 100 Health Points (HP).

A player dies when their HP is reduced to 0, and the kill is awarded to the player who deals the final point of damage.

Multiple players may damage the same enemy, and a player may damage multiple enemies.

The given damage values may be distributed among the enemy team in any valid way that is consistent with these rules.

For every player, determine the minimum and maximum possible number of kills they could have achieved over all valid damage distributions.

You are given an array $$$a$$$ of length $$$n$$$.

A permutation $$$p$$$ of length $$$n$$$ is called good if for each $$$i$$$ from $$$1$$$ to $$$n$$$ the following condition holds:

• Let $$$x$$$ be the number of indices $$$j$$$ such that $$$i \lt j \le n$$$ and $$$p_i \lt p_j$$$, $$$x$$$ must be a multiple of $$$a_{p_i}$$$.

Find the number of good permutations modulo $$$10^9 + 7$$$.

You are given two arrays $$$a$$$ and $$$b$$$ both of length $$$n$$$, the score of a range $$$[l, r]$$$ is defined as follows :

• for each $$$i$$$ from $$$l$$$ to $$$r$$$ add $$$b_i$$$ to the score if $$$a_i = \min\limits_{l \le j \le i} a_j$$$ or $$$a_i = \min\limits_{i \le j \le r} a_j$$$ .

i.e the score of a range is the sum of $$$b_i$$$ over all $$$i$$$ such that $$$a_i$$$ is a prefix minimum or a suffix minimum in the range.

Find the maximum score of a range.

On July 8, 2026, the Homs CPC contest was supposed to take place in Homs.

The judges were very excited about the contest. They had already packed their bags and booked a Pullman bus from Aleppo to Homs to attend the event.

However, the contest was canceled.

One of the judges, Chief Judge Apra, was furious. He had spent nearly two months preparing the problems, and he did not want all that work to go to waste.

So, Apra decided to publish his problems as a Codeforces contest instead.

The contestants from Homs University heard about Apra's plan. They wanted to prevent him from publishing the contest before anyone could read the problems.

They knew that Apra usually uses the password "password".

Your team has reached the login page and wants to guess Chef Apra's password.

Can you predict Chief Apra's password?

$$$Nowar$$$ and $$$Abodeh$$$ are organizing their graduation party. To make the place look more festive, they decided to prepare a large rectangular decoration board of size $$$n \times m$$$.

They have collected several decorative pieces that can be placed on the board:

• $$$s$$$ square tiles of size $$$1 \times 1$$$;
• $$$r$$$ rectangular tiles of size $$$1 \times 2$$$;
• $$$t$$$ right triangular tiles with legs of length $$$1$$$ and $$$1$$$.

Tiles may be rotated by any multiple of $$$90^\circ$$$. Each tile may be used at most once.

$$$Nowar$$$ has a specific design in mind. He wants the decorations to cover exactly a fraction $$$\frac{a}{b}$$$ of the total board area. $$$Abodeh$$$ is not sure whether this can be achieved using the available tiles, so he asks for your help.

No two tiles may overlap.

Determine whether it is possible to choose and place some of the available tiles so that the total covered area is exactly $$$\frac{a}{b}$$$ of the board's area.

Apraham and Nowar are two world-renowned cryptographers who have dedicated their lives to solving the world's most complex puzzles.

One day, they received a mysterious locked briefcase containing a digital screen. On the screen, there was a long sequence of $$$n$$$ numbers where $$$n$$$ is a power of $$$2$$$, represented as an array $$$a$$$ of length $$$n$$$, with indices ranging from $$$0$$$ to $$$n-1$$$.

To unlock the briefcase and reveal the secret inside, they must find all the hidden connections within the sequence. The lock's security system evaluates pairs of indices, $$$i$$$ and $$$j$$$, using two fundamental bitwise operations. For any chosen pair, it calculates the Common Factor (using Bitwise AND, $$$i \ \& \ j$$$) and the Difference Factor (using Bitwise XOR, $$$i \oplus j$$$).

'Apraham notices a small hint engraved on the back of the briefcase: "True balance is achieved only when the paths of intersection and divergence hold the exact same value.'This means a pair of indices $$$(i, j)$$$ is considered a Valid Key Pair if and only if:$$$$$$a[i \ \& \ j] = a[i \oplus j]$$$$$$Nowar needs to find the total number of these Valid Key Pairs $$$(i, j)$$$ where $$$0 \le i, j \lt n$$$ to generate the final override code and open the briefcase.

Can you help Apraham and Nowar calculate the total number of valid pairs?

The year is 2026, and the competitive programming scene is at its absolute peak. Homs-CPC and SVU-CPC are facing each other in a special challenge supervised by the legendary judge Kira and the strict coach Habbab.

To prepare for the challenge, the contestants are arranged in a fixed order. Each contestant has an integer $$$a_i$$$, representing their problem-solving skill level.

Coach Habbab wants to divide the contestants into exactly two distinct strategic groups during a crucial part of the contest: the Main Team and the Support Team.

You are given an array $$$a$$$ of $$$n$$$ integers, representing the skill levels of the contestants. Coach Habbab decides to choose a contiguous subarray from index $$$l$$$ to $$$r$$$ to form the Main Team. The remaining contestants (from index $$$1$$$ to $$$l-1$$$, and from $$$r+1$$$ to $$$n$$$) will form the Support Team. Due to his custom strategy, Coach Habbab must choose the indices such that $$$l \le r$$$ ($$$1 \le l \le r \le n$$$).

The "Judges' Score" of this strategy is calculated as the sum of two values:

1. The maximum skill level among the Main Team:

$$$$$$ \max(a[l], a[l+1], \dots, a[r]). $$$$$$

2. The MEX (Minimum Excluded value) of the skill levels among the Support Team:

$$$$$$ \operatorname{MEX}(a[1],a[2], \dots, a[l-1], a[r+1], a[r+2], \dots, a[n]). $$$$$$

(Note: The MEX of a set of integers is the smallest non-negative integer that does not belong to the set).

Kira and Coach Habbab want to find the best possible strategy that minimizes this total Judges' Score.

Help the contestants of Homs-CPC and SVU-CPC find the minimum possible score by choosing the optimal subarray $$$(l, r)$$$ under the condition $$$l \le r$$$.

You are given a tree consisting of $$$N$$$ vertices numbered from $$$1$$$ to $$$N$$$. The tree has $$$N-1$$$ weighted edges, where the $$$i$$$-th edge connects vertices $$$u_i$$$ and $$$v_i$$$ and has a weight $$$w_i$$$ ($$$1 \le w_i \le M$$$).

Let $$$S$$$ be the set of all simple paths in the tree that contain at least one edge. A simple path can be uniquely identified by its unordered pair of distinct endpoints $$$(u, v)$$$ ($$$1 \le u \lt v \le N$$$). Thus, there are exactly $$$|S| = \frac{N(N-1)}{2}$$$ paths in the set $$$S$$$. Each path $$$x \in S$$$ can be viewed as a set of edges.

For any two paths $$$x, y \in S$$$, we denote $$$x \cap y$$$ as the set of edges that are common to both paths. We define a cost function $$$f(x, y)$$$ as follows:

$$$$$$f(x, y) = ( \sum_{e \in x \cap y} w_e ) \cdot \gcd_{e \in x \cap y}(w_e)$$$$$$

I.e.: the value of a function is the sum of the weights on the intersection path of $$$x$$$ and $$$y$$$ times the GCD of the weights on the intersection path of $$$x$$$ and $$$y$$$, where $$$\gcd(i, j)$$$ is the greatest common divisor of $$$i$$$ and $$$j$$$.

Your task is to calculate the total sum of $$$f(x, y)$$$ over all ordered pairs of paths $$$(x, y)$$$ from the set $$$S$$$.

Since the total sum can be very large, output it modulo $$$998244353$$$.

$$$Abodeh$$$, $$$Nowar$$$, and $$$Hussein$$$ decided to organize a football tournament. In each match, exactly $$$2$$$ players are required to play. For any village with $$$x$$$ people, they want to know how many full matches they can organize, leaving at most one person as a substitute. Let this number be $$$f(x)$$$.

Now, they are given a range of villages from $$$L$$$ to $$$R$$$. They need your help to calculate the total number of matches they can organize across all these villages.

More formally:

$$$f(x)$$$ is the number of terms equal to $$$2$$$ in the representation of $$$x$$$ as a sum of several $$$2$$$'s and, if needed, one $$$1$$$.

For example:

• $$$5 = 2 + 2 + 1 \rightarrow f(5) = 2$$$.
• $$$6 = 2 + 2 + 2 \rightarrow f(6) = 3$$$.
• $$$7 = 2 + 2 + 2 + 1 \rightarrow f(7) = 3$$$.

You need the sum of $$$f(x)$$$ for all $$$x$$$ from $$$L$$$ to $$$R$$$:

$$$$$$ Answer = \sum_{x=L}^{R} f(x). $$$$$$

Edward and Sawaha are two brilliant scientists working at a secret data research facility.

Recently, they uncovered two ancient digital archives containing encrypted energy frequencies.

For security reasons, both archives contain an identical number of data elements. Edward manages the first archive, which can be represented as an array $$$A$$$, while Sawaha is in charge of the second archive, represented as an array $$$B$$$.

To unlock the core secret of these frequencies, Edward and Sawaha need to synchronize the data by forming pairs.

A pair is created by matching exactly one element from Edward's archive with exactly one element from Sawaha's archive. Each element from either archive can be used at most once, meaning that no element can belong to more than one pair.

The power value of a synchronized pair containing the elements $$$a$$$ and $$$b$$$ is calculated as their product ($$$a \times b$$$). The entire system will stabilize only if they can select a certain number of pairs such that every single selected pair has a power value greater than or equal to the total number of selected pairs.Edward and Sawaha want to achieve the maximum possible level of system stability.

Your task is to help them determine the maximum integer $$$x$$$ such that it is possible to choose at least $$$x$$$ completely disjoint pairs, where the product of the elements in each chosen pair is at least $$$x$$$.

Edward is shopping in Homs. There are $$$n$$$ shops arranged in a row, and shop $$$i$$$ sells one item with price $$$a_i$$$.

The street has two entrances:

• the left entrance, located before shop $$$1$$$;
• the right entrance, located after shop $$$n$$$.

Initially, Edward stands outside the left entrance and has $$$k$$$ money.

Edward starts from the left entrance. He may buy a prefix of items:

$$$$$$ \left[a_1, a_2, \ldots, a_x\right] $$$$$$

where $$$0 \le x \le n$$$. The prefix is allowed to be empty.

After that, Edward may switch entrances at most once. If he switches, he enters from the right entrance and buys a suffix of items in the following order:

$$$$$$ \left[a_n, a_{n-1}, \ldots, a_{n-y+1}\right] $$$$$$

where $$$0 \le y \le n-x$$$. The suffix is allowed to be empty.

Therefore, for some $$$x$$$ and $$$y$$$, Edward buys items in the exact order:

$$$$$$ \left[a_1, a_2, \ldots, a_x, a_n, a_{n-1}, \ldots, a_{n-y+1}\right] $$$$$$

The shop owner gives Edward $$$d$$$ coupons.

Before buying an item, Edward may use one unused coupon on it. If he does, the item's price becomes $$$0$$$, and Edward can buy it without spending any money.

Each coupon can be used on at most one item, and each item can be bought at most once.

Before buying an item without using a coupon, Edward must have at least its price. After buying it, his amount of money decreases by its price. Note that if an item's price is negative, buying it increases Edward's money.

Find the maximum number of items Edward can buy without ever being unable to afford an item.

You are given an integer $$$M$$$ and you have an empty array $$$a$$$. You have to process $$$q$$$ queries on array $$$a$$$, there are two types of queries :

• $$$1$$$ $$$x$$$ $$$k$$$ $$$(1 \le x \le 10^9, \ 2 \le k \le M)$$$ : insert $$$x$$$ at positions $$$1, k + 1, 2 * k + 1, \dots ,\left\lfloor \frac{length(a)}{k - 1} \right\rfloor * k + 1$$$, where $$$length(a)$$$ is the length of array $$$a$$$ before the inserting the number. for example if the array was $$$[1,1,1,1,1,1,1]$$$ and the query was $$$1$$$ $$$2$$$ $$$3$$$, the array will become : $$$[2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1]$$$.
• $$$2$$$ : print the first element of the array and then delete it.

Majed is working on an old digital calculator. Unfortunately, some of its displays are damaged, and each displayed number contains exactly one corrupted character instead of a digit.

The calculator shows two strings $$$x$$$ and $$$y$$$, each of length exactly $$$3$$$. Every string contains exactly two decimal digits ('0' to '9') and one non-digit character. The corrupted character may appear in any position.

To recover the original numbers, Majed removes the corrupted character from each string. The remaining digits keep their relative order and form a two-digit integer. It is guaranteed that the resulting integers do not contain leading zeros.

Help Majed determine the product of the two recovered integers.

Coach $$$Apraham$$$ informed $$$Kira$$$ that the Homs-CPC would be held soon and requested that they

draft some competition problems.

$$$Kira$$$ proposed this particular problem, considering it to be an easy task, and now requests your solution.

You are given a non-negative integer $$$n$$$.

Count the number of non-negative integers $$$x$$$ such that :

• $$$n + x = n \oplus x $$$, where $$$\oplus$$$ denotes bitwise XOR.
• $$$x$$$ ($$$0 \le x \le 2^{30}$$$ -$$$1$$$)