This is a different version of problem B. The only difference between the two versions is the validity condition for a coloring.

After competing virtually in all CodeForces rounds of the past three years, Geosé finally mastered graph problems. Graph coloring became his favorite topic in graph theory. While looking for a problem that combines this topic and geometry, he came up with the following problem:

You are given a convex polygon with $$$n$$$ vertices. For each pair of vertices, a line segment connecting them is drawn. Note that $$$\binom{n}{2}$$$ segments are drawn in total.

You must assign a color to each segment. A coloring is said to be valid if it satisfies the following condition:

• If two segments intersect, they must have different colors.

Two segments are considered to intersect if they share at least one common point, including endpoints. In particular, segments that share a vertex are also considered intersecting.

Determine the minimum number of colors required to obtain a valid coloring.

Note: since the input and output are large, you may need to optimize them for this problem.

For example, in C++, it is enough to use the following lines at the start of the main() function:

int main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr); std::cout.tie(nullptr);
}

This is a different version of problem A. The only difference between the two versions is the validity condition for a coloring.

After competing virtually in all CodeForces rounds of the past three years, Geosé finally mastered graph problems. Graph coloring became his favorite topic in graph theory. While looking for a problem that combines this topic and geometry, he came up with the following problem:

You are given a convex polygon with $$$n$$$ vertices. For each pair of vertices, a line segment connecting them is drawn. Note that $$$\binom{n}{2}$$$ segments are drawn in total.

You must assign a color to each segment. A coloring is said to be valid if it satisfies the following condition:

• If two segments do not intersect, they must have different colors.

Two segments are considered to intersect if they share at least one common point, including endpoints. In particular, segments that share a vertex are also considered intersecting.

Determine the minimum number of colors required to obtain a valid coloring.

Note: since the input and output are large, you may need to optimize them for this problem.

For example, in C++, it is enough to use the following lines at the start of the main() function:

int main() {
    std::ios::sync_with_stdio(false);
    std::cin.tie(nullptr); std::cout.tie(nullptr);
}

Barnaby Nickelsworth III is a peculiar fellow with an even more peculiar hobby: he insists on calling all his friends by nicknames. However, Barnaby's definition of a nickname is quite specific, he only uses prefixes of their actual names.

For instance, his friend Roberto can be called Rob, Robe, Rober, Robert, or even the full Roberto. Barnaby finds endless amusement in cycling through all possible prefix-nicknames for each friend.

However, Barnaby believes that overly short nicknames lose their charm. Calling someone just "R" feels lazy and uninspired. Therefore, he has established a rule: **valid nickname must have a length of at least K characters.

Recently, Barnaby discovered a troubling issue. Some of his friends share common prefixes! For example, among Francisco, Franz, and Frank, all three share the prefix Fran. This means Barnaby might accidentally call out "Fran!" and have multiple friends respond, an unacceptable social catastrophe.

Being the dramatic person he is, Barnaby has decided to end friendships until every valid nickname (prefix of length at least K) belongs to exactly one remaining friend. No two remaining friends should share any prefix of length $$$\geq K$$$.

Help Barnaby determine the maximum number of friends he can keep while satisfying his obsessive nickname uniqueness requirement.

You are given a prime number $$$p$$$ and a matrix $$$A$$$ of size $$$n \times m$$$. Rows are numbered from $$$1$$$ to $$$n$$$ and columns are numbered from $$$1$$$ to $$$m$$$. We will denote the element at the intersection of the $$$i$$$-th row and the $$$j$$$-th column as $$$a_{i,j}$$$. Every element of the array $$$A$$$ is an integer between $$$1$$$ and $$$p-1$$$.

We call a matrix beautiful if it satisfies the following condition:

• $$$p$$$ divides the determinant of every non-trivial square submatrix of $$$A$$$.

Note that a matrix with no non-trivial square submatrix is beautiful.

A submatrix is defined by four integers $$$(i_1,j_1,i_2,j_2)$$$, where $$$1 \leq i_1 \leq i_2 \leq n \text{ and } 1 \leq j_1 \leq j_2 \leq m$$$, and consists of all elements $$$a_{i,j}$$$ of this matrix such that $$$i_1 \leq i \leq i_2$$$ and $$$j_1 \leq j \leq j_2$$$. A submatrix is called non-trivial if $$$i_1 \lt i_2$$$ and $$$j_1 \lt j_2$$$. A submatrix is called square if $$$i_2-i_1 = j_2-j_1$$$.

Find the number of submatrices of $$$A$$$ that are beautiful.

This summer, ESCOM willl hold the XVIII Annual Drone Kaleidoscope (DK for short). During this event, a one-lane track is written as a string.

During the annual DK, several teams, one by one, perform a show as follows:

• Each team picks a different palindrome that actually appears somewhere on the track.
• Each team selects some subset of occurrences of their chosen palindrome, and places a drone in the leftmost character of the selected occurrences.
• To avoid radio noise within the team performance, that team's drone starting positions must be at least $$$k$$$ characters apart.
• Finally, the team flies their drones over the selected occurrences of the chosen palindrome.

A team's show value is $$$$$$ \text{(length of its palindrome)} \times \text{(number of drones it places)}. $$$$$$

For example, if we have $$$s=abaaba$$$ and $$$k=3$$$, a valid show configuration is to field $$$3$$$ teams as follows:

1. The first team chooses the palindrome $$$aba$$$ and launches $$$2$$$ drones with initial positions $$$\{1,4\}$$$. This team's show value is $$$3 \times 2=6$$$.
2. The second team chooses the palindrome $$$baab$$$ and launches $$$1$$$ drones with initial position $$$\{2\}$$$. This team's show value is $$$4 \times 1=4$$$.
3. The third team chooses the palindrome $$$a$$$ and launches $$$2$$$ drones with initial positions $$$\{1,6\}$$$. This team's show value is $$$1 \times 2=2$$$. Note that launching $$$3$$$ drones with initial positions $$$\{1,3,6\}$$$ is invalid, since starting positions should be at least $$$k=3$$$ characters apart.

The total show value in this example is $$$6+4+2=12$$$.

Your team is practicing to participate in this event. To practice flying a drone over a palindrome, your team needs to write a palindrome consisting only of lowercase letters. Given an integer $$$n$$$, print a palindrome of length $$$n$$$.

Recall that a palindrome is a string that reads the same backward as forward, for example, strings "z", "aba", "abccba" are palindromes, but strings "codeforces", "reality", "ab" are not.

Danna is organizing a downhill tournament called "The Montes Tournament" on a peculiar mountain.

The mountain consists of $$$N - 1$$$ checkpoints, numbered from $$$2$$$ to $$$N$$$. Each checkpoint is connected by directed paths following a strict rule:

For each checkpoint $$$v$$$, there exists a directed edge $$$v → u$$$ if and only if $$$u$$$ divides $$$v$$$.

Note that the checkpoint with value $$$1$$$ does not exist.

A player starts at a checkpoint $$$x$$$ and wins if they manage to reach a target checkpoint $$$y$$$. However, Danna knows that one of the players is a cheater.

Although the cheater can bend the rules, they have a serious weakness.

The cheater has never seen any checkpoint higher than their starting position $$$x$$$. The thought of climbing to an unknown height terrifies them.

Because of this fear, the cheater will never move to any checkpoint $$$z$$$ such that $$$z \gt x$$$, under any circumstances.

This restriction applies even when cheating.

The cheater may perform at most one cheating move during the entire run.

A cheating move allows the player to take an edge that is originally directed into their current checkpoint and move to its origin.

Example:

There exists a directed edge $$$4 \rightarrow 2$$$. If the player is currently at checkpoint $$$2$$$, they may cheat to move to checkpoint $$$4$$$.

Once the cheating move is used, the cheater stop moving.

You are given $$$Q$$$ independent queries.

For each query, you are given a starting checkpoint $$$x$$$. A target checkpoint $$$y$$$ is chosen uniformly at random from all integers such that: $$$2 \leq y \leq N$$$.

Your task is to compute the probability that the cheater cannot reach $$$y$$$ starting from $$$x$$$, using directed edges and at most one cheating move.

After the ICPC World Finals in Luxor, the team Exceso de Chilaquiles explored the Mortuary Temple of Hatshepsut. Wandering through the ancient ruins, they discovered a mysterious stairway hidden beneath the temple.

The stairway contains $$$N+10^{18}+1$$$ steps numbered from $$$-10^{18}$$$ to $$$N$$$ from the lowest step to the highest one. The non-positive steps are underground, and the positive ones are over the surface.

Step $$$i$$$ is carved with an ancient integer $$$a_i$$$. According to the inscriptions, the integer of every step points to a lower step where the stone is more stable. For this reason, from step $$$i$$$ you are forced to jump to step $$$a_i$$$, where $$$a_i \lt i$$$.

The stairway is unstable. Every time someone jumps from a step, the stone cracks slightly, reducing its written integer by a value $$$k$$$. The value of $$$k$$$ may vary over time. More formally, after jumping from step $$$i$$$, the value of $$$a_i$$$ is reduced by a given value $$$k$$$.

The team has $$$Q$$$ plans to walk the stairway. During each plan, they will start at the step $$$s$$$ and jump through multiple steps until reaching a step $$$i$$$ such that $$$i \leq 0$$$ (they don't want to go too far underground). The team knows the value of $$$k$$$ for each of their plans.

The team is mentally exhausted from solving the problems of the contest, so they need your help to find the number of the step where they arrive at for each plan.

Note that all changes applied to the values $$$a_i$$$ during a plan remain for all subsequent plans.

You are chilling during your vacations, simulating divs and playing lol, when Nijika asks you a strange question.

Suppose an island shaped like a convex polygon undergoes morphological dilation using a disk of radius $$$r$$$ (resulting in a uniform expansion in all directions). What is the minimum radius $$$r$$$ such that the island reaches an area of at least $$$k$$$?

Formally, you are asked to find the minimum radius $$$r$$$ such that the Minkowski sum of the Polygon $$$P$$$ and a disk $$$D$$$ of radius $$$r$$$ results in an area greater than or equal to $$$k$$$.

The first thing that crossed your mind was: "Am I invited to this island?"

The answer is a resounding no (bro?). Now, help Nijika with her homework.

You are given a system of $$$N$$$ underground burrows connected by $$$M$$$ directed tunnels. The burrows are numbered from $$$1$$$ to $$$N$$$. The tunnels are one-way to avoid congestion. The only entrance from the surface is burrow $$$1$$$, and every burrow is reachable from burrow $$$1$$$.

Ferrets live in these burrows. They plan to choose some burrows as sleeping burrows and one additional burrow as an emergency supply storage. For safety, once the supplies are placed, ferrets are not allowed to enter the storage burrow, so any route that goes through that burrow cannot be used.

The ferrets want to simulate $$$Q$$$ different scenarios. In each scenario, you are given $$$k$$$ burrows that would serve as sleeping burrows. Your task is to determine how many burrows could be chosen as the supply storage burrow such that it is still possible to travel from burrow $$$1$$$ to all sleeping burrows without passing through the chosen supply storage burrow. Note that it is not possible to select burrow $$$1$$$ or any sleeping burrow as the supply storage burrow.

Help the ferrets find the answer for all scenarios.

You're playing Fallout Van Buren, an unreleased post-apocalyptic RPG. In this version, character builds are simplified to two stats: Strength (S) and Armor (E). Strength determines your damage output, while Armor represents your Power Armor's integrity.

You're about to face N hostile creatures in the wasteland. Each enemy i has damage $$$A_i$$$ and armor $$$D_i$$$.

Combat Mechanics:

• Combat is turn-based; you always strike first

• Each attack reduces the target's armor by the attacker's Strength

• A combatant is defeated when they receive an attack while their armor is already at zero — think of it as your Power Armor absorbing hits until it breaks, then the next hit is fatal

• Your Power Armor fully repairs between encounters

• Stats are assigned once and cannot be changed

Find the minimum $$$X$$$ such that you can defeat all $$$N$$$ enemies.

You are given a dictionary of $$$n$$$ strings $$$s_1, s_2, \dots, s_n$$$.

We define $$$f(t)$$$ as the number of ways of obtaining a string $$$t$$$ by the following three-step operation:

1. Select two (not necessarily distinct) indices $$$i$$$ and $$$j$$$ from the dictionary.
2. Select an integer $$$k$$$ such that $$$0 \leq k \leq |s_i|$$$. Let $$$a$$$ be the prefix of $$$s_i$$$ of length $$$k$$$, and let $$$b$$$ be the suffix of $$$s_i$$$ of length $$$|s_i|-k$$$. $$$a$$$ or $$$b$$$ may be empty.
3. Concatenate the strings $$$a + s_j + b$$$. If this concatenation is equal to $$$t$$$, it counts as one valid way to obtain $$$t$$$.

Each valid triple $$$(i, j, k)$$$ is counted separately.

You are given $$$q$$$ strings $$$t_1, t_2, \dots, t_q$$$. For each $$$t_i$$$ ($$$1 \leq i \leq q$$$), find the value of $$$f(t_i)$$$.