Count all pairs of integers $$$(n, p)$$$ such that $$$0 \le p \le P$$$, $$$p \neq n \cdot p$$$, and $$$p!=n \cdot p$$$.

Yash really likes monkeys and special arrays. Because he was curious, Curios George gave Yash an array $$$A$$$ of length $$$N$$$, and now Yash wonders how many good subarrays exist in this array.

Specifically, Yash deems a subarray good if $$$X$$$, his favorite number, is the maximum value in the subarray, $$$Y$$$, his second favorite number, is the minimum in the subarray, and $$$K$$$, his least favorite number, is not in the subarray. In other words, in a given subarray, $$$X$$$ should be the maximum value, $$$Y$$$ should be the minimum, and $$$K$$$ should not be in the subarray.

There is a chain of $$$n$$$ nodes with edges between nodes $$$u$$$ and $$$v$$$ such that $$$|v - u| = 1$$$. For each node $$$i$$$ ($$$1 \le i \le n$$$), you will keep it with probability $$$\frac{1}{i}$$$ or delete it with probability $$$1 - \frac{1}{i}$$$. Find the expected number of connected components after all the deletions, modulo $$$10^9+7$$$.

Recently, two very productive (even peakly productive) Rinbot'ers have been optimizing Rinbot. They realized that instead of buying a sentient supercomputer, they could use hangman strats instead. These two highly productive Rinbot'ers made a giant cloud-hosted database where they stored the title for every anime in Rinbot. Unfortunately, the AC turned off, marking the end of the laptop.

Can you write a program to access their database?

In this link, there is a text file with all anime titles for shows in Rinbot (up until the Spring 2024 season).

For $$$T$$$ titles, determine which line in the file the title is modulo $$$P$$$. For instance, "86" is on line $$$75$$$ and "?" is on line $$$5$$$.

The Teamscode Online Judge has a submission size limit of 50 kB.

Jason wants to orz Waymo! To achieve this, he can use the following operations:

1. Type "orz" in $$$A$$$ seconds.

2. Copy all the text in $$$B$$$ seconds.

3. Paste all the copied text in $$$C$$$ seconds.

Since Jason is in a hurry, please help him minimize the amount of time he spends to orz Waymo at least $$$N$$$ times.

You are given a value $$$x$$$, a permutation $$$p$$$ of length $$$n$$$, and an array $$$q$$$ of length $$$n$$$. For every $$$p_i$$$, you must choose to either set $$$x := x + p_i$$$ or set $$$x := x \oplus 2^{d(p_i)}$$$, where $$$\oplus$$$ denotes the bitwise XOR operation and $$$d(a)$$$ denotes the sum of the digits of $$$a$$$ in its decimal representation. For each $$$k$$$ ($$$1 \le k \le n$$$), find the number of possible values of $$$x$$$ that satisfy $$$x \leq q_k$$$ after processing the elements $$$p_1, p_2, \ldots, p_k$$$ in order from $$$1$$$ to $$$k$$$.

The Rhodes Island operators are scouting out the new map for CCB#2. The map can be represented as a tree with $$$n$$$ nodes numbered from $$$1$$$ to $$$n$$$. Slugs of $$$m$$$ different colors exist, and there is one slug of color $$$i$$$ ($$$1 \le i \le m$$$) for each node on the shortest path from $$$s_i$$$ to $$$t_i$$$. Note that multiple slugs of different colors can be on the same node, and not all colors of slugs are initially on the map.

The slugs have a very particular behavior pattern where sometimes all the slugs of the same color will leave the map, or all the slugs of the same color will return to the map.

Ifrit is tasked with clearing out the slugs, so she will consider different plans of attack. A plan of attack is denoted by two nodes $$$u$$$ and $$$v$$$ in which she hits all the slugs on the shortest path from $$$u$$$ to $$$v$$$. If she hits any slug with color $$$i$$$, she adds $$$v_i$$$ to her score, where $$$v_i$$$ is a predetermined value for color $$$i$$$. Ifrit will not actually perform the attack, but she wants to know her score for each plan of attack.

You will have to process $$$q$$$ events, each one of the following three types:

1. All the slugs of color $$$i$$$ return to their respective nodes on the tree.
2. All the slugs of color $$$i$$$ leave the tree.
3. Calculate the score Ifrit would achieve if she performed an attack by selecting nodes $$$u$$$ and $$$v$$$.

Thomas has recently begun realizing his dream of being the best mangaka in the world. He sees his future in the equation $$$E = Mc^2 + \text{AI}$$$: $$$\text{Events} = \text{Main character}^2 + \text{love}$$$.

Thomas first asks ChatGPT to generate $$$n$$$ ($$$n$$$ is even) character tropes. Then he makes a matrix $$$a$$$ of size $$$n$$$ by $$$n$$$, where $$$a_{ij}$$$ denotes the tension of someone of trope $$$i$$$ having feelings for someone of trope $$$j$$$. $$$a_{ij} = -1$$$ if the trope $$$i$$$ is unable to have any feelings for trope $$$j$$$. Note that $$$a_{ij}$$$ is not necessarily equal to $$$a_{ji} $$$.

Thomas will assign half of his $$$n$$$ tropes to male characters and the other half to female characters. He will draw feelings from each male to some female such that no two males have feelings for the same female. Then he will draw feelings from each female to each male such that no two females have feelings for the same male. Note that feelings are not necessarily reciprocated.

Note that feelings are not necessarily reciprocated. Let $$$p_i$$$ denote the trope for which the $$$i$$$-th trope has feelings. The value of this assignment (of gender and feelings) is

$$$$$$ \frac{\prod_{i \text{ is male}} a_{ip_i}}{\prod_{i \text{ is female}} a_{ip_i}}$$$$$$

Find the sum of values over all possible assignments of gender and feelings. Since this number may be very large, please find it modulo $$$998244353$$$.

Noko-tan is playing the hit game "Flappy Deer". In this game, Noko-tan controls a flying deer who moves in an infinite 2-D grid. The deer is initially located at some cell. Every second, the deer will move to the right by a single unit (increase $$$x$$$ coordinate by one), and Noko-tan can choose whether to move the deer up by one unit (increase $$$y$$$ coordinate by one), keep the deer's current height, or move the deer down by one unit (decrease $$$y$$$ coordinate by one). There are also $$$N$$$ full water bottles that Noko-tan is deathly afraid of. Thus, if her flying deer ever shares a cell with a water bottle, the game will immediately end. The $$$i$$$-th water bottle is a rectangle occupying the cells in between $$$(x_i, a_i)$$$ and $$$(x_i, b_i)$$$. Finally, to make things interesting, there are also deer crackers floating around. The $$$j$$$-th is located at cell $$$(x_j, y_j)$$$. Noko-tan collects a deer cracker if her flying deer occupies the same cell as that deer cracker. Since Noko-tan has antlers for brains, she doesn't know how to play in order to collect the maximum number of deer crackers. Thus, she asks you to help her find this value for $$$Q$$$ starting positions. Note that Noko-tan encountering a water bottle ends the game, but does not reset the number of crackers she has collected. Also, the grid is infinite, so there is no bounds on Noko-tan's maximum $$$x$$$ or $$$y$$$ coordinates.

You are given a rectangular grid $$$B$$$ with $$$N$$$ rows and $$$M$$$ columns. Consider some rectangular grid $$$A$$$ with $$$N$$$ rows and $$$M$$$ columns. $$$A$$$ is considered mediocre if $$$A_{i, j}$$$ is an integer and $$$0 \leq A_{i, j} \leq B_{i, j}$$$ for all $$$(i, j)$$$.

Let $$$F(A) = \displaystyle\left(\prod_{i = 1}^{N}\left(\sum_{j = 1}^{M}A_{i, j}\right)\right) \displaystyle\left(\prod_{j = 1}^{M}\left(\sum_{i = 1}^{N}A_{i, j}\right)\right)$$$.

Calculate the sum of $$$F(A)$$$ over all mediocre $$$A$$$ modulo $$$998244353$$$.

The Astral Express is a special train that traverses the galaxy. The galaxy initially consists of $$$2 \cdot n$$$ planets represented as an array $$$a_1, a_2, \ldots, a_{2 \cdot n}$$$, where the $$$i$$$-th planet initially has a gravitational coefficient of $$$a_i = 1$$$ if $$$i \le n$$$ and $$$a_i = -1$$$ if $$$i \gt n$$$.

The Astral Express moves in steps, and the direction of the movement is determined by its velocity $$$v$$$. At the beginning of each step, if $$$v \gt 0$$$, then the Express moves one index to the right. If $$$v \lt 0$$$, then the Express moves one index to the left. If $$$v = 0$$$, then the crew moves in the direction they moved previously. At the end of each step, if the Express is at position $$$i$$$, then $$$a_i$$$ is added to $$$v$$$. If the Astral Express goes past the bounds of the array, then its journey will end.

Since the galaxy is unstable, the planets are constantly breaking apart and merging. A planet with $$$|a_i| = 2$$$ may split into two adjacent planets with gravitational coefficients of $$$\frac{a_i}{2}$$$, and two identical adjacent planets with $$$a_i = a_{i + 1}$$$ and $$$|a_i|= 1$$$ may merge into a single planet with gravitational coefficient $$$a_i + a_{i + 1}$$$.

The Astral Express crew is curious about how their train will move in this unstable galaxy, so you are given $$$q$$$ queries of $$$3$$$ types:

• "1 x" — The Astral Express begins moving at the rightmost positive $$$a_i$$$ with their given initial velocity equal to $$$x$$$. Output how many steps the Astral Express will take before their journey ends, or output $$$-1$$$ if their journey will go on forever.
• "2 x" — Merge $$$a_x$$$ and $$$a_{x + 1}$$$ into a new planet with gravitational coefficient $$$a_x + a_{x + 1}$$$. Formally, set $$$a_x$$$ to $$$a_x + a_{x + 1}$$$, and remove $$$a_{x + 1}$$$ from the array. Note that the indices of all elements right of $$$a_{x + 1}$$$ will decrease by one. It is guaranteed that $$$a_x = a_{x + 1}$$$ and $$$|a_x| = 1$$$.
• "3 x" — Split $$$a_x$$$ into two new planets both with gravitational coefficients $$$\frac{a_x}{2}$$$. Formally, set $$$a_x$$$ to $$$\frac{a_x}{2}$$$ and insert a new planet with gravitational coefficient $$$a_x$$$ at index $$$x + 1$$$. Note that the indices to the right of the inserted element will increase by one. It is guaranteed that $$$|a_x| = 2$$$.

For example, given $$$n = 3$$$, our initial array will look like $$$[1, 1, 1, -1, -1, -1]$$$, and we can perform an operation "2 2", merging $$$a_2$$$ and $$$a_3$$$ together, resulting in an array of $$$[1, 2, -1, -1, -1]$$$. We can then perform an operation "3 2", splitting $$$a_2$$$ into two elements, resulting in an array of $$$[1, 1, 1, -1, -1, -1]$$$.