Name any show which is referenced in this contest.

Goma has recently developed a newfound interest in mountain climbing and is traveling from left to right through a path of length $$$N$$$ with altitudes assigned to each point. Your task is to determine the number of mountains that Goma will climb during their journey. A mountain is defined as three or more consecutive points with strictly increasing altitudes. It is important to note that if there are more than three points with strictly increasing altitudes, they should be considered as a single mountain. For example, if $$$[l, r]$$$ is a mountain, $$$[l, r-1]$$$ should not be considered as a separate mountain.

As is a customary tradition of very hardworking, diligent, productive, and busy problemsetters, Thomas (and other members of the Teamscode problemsetting team) like to play rinbot after meetings.

In a game of rinbot, there are $$$n$$$ rounds and $$$k$$$ problemsetters playing along with Thomas. There is always exactly $$$1$$$ winner per round, and a game is considered a sweep if Thomas is the winner in every single round.

Waymo tells you the conditions of the game and wants to know how many ways different problemsetters can win each round such that Thomas does not get a sweep.

Chisato has an $$$N \times N$$$ multiplication table. A multiplication table is an $$$N \times N$$$ matrix where on cell $$$(i, j)$$$ the number $$$i \cdot j$$$ is written for all $$$i \in [{1 \cdots N}]$$$ and $$$j \in [{1 \cdots N}]$$$.

A $$$3 \times 3$$$ mulitplication table looks like this:

$$$$$$ 1, 2, 3 \\ 2, 4, 6 \\ 3, 6, 9 $$$$$$

Takina challenges her to find the smallest positive number that is not present in the multiplication table. Chisato knows how to find the answer, but do you?

Thomas has two arrays $$$a$$$ and $$$b$$$ both of length $$$n$$$, as well as a value $$$k$$$. He can perform the following operations on $$$a$$$:

1. Choose an index $$$1\leq i\leq n$$$, and set $$$a_i:=a_i+1$$$
2. Choose an index $$$1\leq i\leq n$$$, and set $$$a_i:=a_i-1$$$
3. Choose an index $$$1\leq i\leq n-k+1$$$, and cyclically shift the subarray $$$[i, i + k - 1]$$$ to the left. This operation can be done at most once.

A cyclic shift on a subarray $$$[l,r]$$$ of array $$$a$$$ assigns $$$a_i:=a_{i+1}$$$ for $$$l\leq i \lt r$$$ and $$$a_r:=a_l$$$.

Help Thomas find the minimum number of operations to make the two arrays equal.

Despite being best girl, Kirisu-sensei still is terrible at math. She has combined the concepts of greatest common factor and least common multiple together!

The greatest common multiple of two numbers $$$a$$$ and $$$b$$$ (gcm) is the largest number that both $$$a$$$ and $$$b$$$ can divide. Since this number is very large (in fact it's infinite), you are asked to find it mod some number $$$c$$$. In this context, largest under mod means the maximum value of $$$(d \text{ mod } c)$$$ over any $$$d$$$ that both $$$a$$$ and $$$b$$$ divide $$$d$$$.

In mathematical terms,

$$$$$$x = \max_{a \mid d, b \mid d}(d \text{ mod } c)$$$$$$

output $$$x$$$ as defined above ($$$a \mid d$$$ means that $$$a$$$ divides $$$d$$$).

You are playing a videogame. In this game, there's an assassin throwing daggers at you, one dagger is thrown every $$$t$$$ seconds and the dagger doesn't take any time to arrive to its destination. You are running through the $$$x$$$-axis of the Euclidean Plane at a constant speed of $$$1$$$ unit per second, and you can decide to stop at any point for some number of seconds. Your goal is to arrive to the coordinate $$$n$$$ in the $$$x$$$-axis without being killed by the assassin.

The game consists of $$$q$$$ levels, each numbered from $$$1$$$ to $$$q$$$, and each level will add one more shield that will be present on this and the next levels. You are safe from a dagger if you are at a shield or at the end point when the dagger is thrown. If not, you will lose. Shields don't vanish when they protect you from a dagger, so you can use the same shield multiple times. Additionally, for each level, $$$t$$$ is equal to the number of that level (for level $$$1$$$, $$$t$$$ will be equal to $$$1$$$; for level $$$2$$$, $$$t$$$ will be equal to $$$2$$$; and so on). Note that this game is a bit weird, so you will always pass to the next level and start again from coordinate $$$0$$$, regardless of whether you have managed to arrive to the coordinate $$$n$$$ or not.

You want to boost your performance so you have decided to calculate if you will be able to pass each level and, if it's possible to do so, the least number of seconds you need to do it.

Misaka has come up with a very good problem for this contest. Unfortunately, that problem already exists (on at least 3 different online judges!) so she has to add a creative twist to it!

Consider a full binary tree of depth $$$d$$$. Remember a full binary tree with depth $$$d$$$ has $$$2^d - 1$$$ nodes and there is an edge from each node $$$u$$$ $$$(2 \leq u \leq 2^d -1)$$$ to its parent $$$\left \lfloor \frac{u}{2} \right \rfloor$$$. Misaka asks you to find the sum of distances between every pair of nodes on the tree.

Formally if $$$n = 2^d - 1$$$ ($$$n = \#$$$ nodes on the tree), compute:

$$$$$$S = \displaystyle\sum_{i = 1}^{n} \sum_{j = 1}^{n} dist(i, j)$$$$$$

where $$$dist(i, j)$$$ is the minimum number of edges on any path between node $$$i$$$ to node $$$j$$$.

Goma has been enjoying mountain climbing so much lately that they want to take Peach with them! There is a path of length $$$N$$$ with varying altitudes, and Peach wants to know how many distinct points they can see from certain spots on the path. However, fog is forecasted for the day of their trip, so Peach wants to take that into account when calculating the visible points. Specifically, if a point $$$p$$$ has altitude $$$a_p$$$ and the fog has altitude $$$f$$$, then Peach can see all points $$$q$$$ such that:

1. The altitude $$$a_q$$$ is strictly less than $$$a_p$$$
2. There are no points between $$$p$$$ and $$$q$$$ with altitude $$$a_p$$$ or greater
3. Either $$$a_q \lt a_p \lt f$$$ or $$$f \leq a_q \lt a_p$$$ (points at or above the fog cannot see points strictly below it)

Note that Peach can always see the spot they are currently on.

Quintessential Quintuplets is a good show but let's be real here, only two of the quintuplets are actually quintessential.

Nino and Miku are playing a game on an array $$$a$$$ of $$$n$$$ positive integers. In one turn, a player can pick any element in the array and floor divide it by $$$2$$$ or $$$3$$$ as long as the resulting number remains strictly positive. Nino goes first and the player who cannot make a move loses.

A possible game on the array $$$a = [1, 2, 5]$$$ could go as follows:

1. Nino divides $$$a_3$$$ by $$$3$$$ and is left with $$$a_3 = \left \lfloor \frac{5}{3} \right \rfloor = 1$$$ with the array now becoming $$$[1, 2, 1]$$$

2. Miku can then divide $$$a_2$$$ by $$$2$$$ leaving the array as $$$[1, 1, 1]$$$. Note that Miku could not have divided by $$$3$$$ since $$$\left \lfloor \frac{2}{3} \right \rfloor = 0$$$ and that would no longer be positive.

3. Nino then loses since she cannot divide any of the remaining elements by any value. In this game, Nino did not play optimally and lost.

When given the array, assuming both girls play perfectly, who will win?

Hina has a string $$$s$$$ of lowercase letters and question marks. She wants to fill each question mark with a lowercase letter so that she is left with a string of only lowercase letters. Hina will then count over all permutations of letters in her string, the number of unique strings that can be formed. For instance, the string $$$aab$$$ has $$$6$$$ permutations, $$$[aab, aba, aab, aba, baa, baa]$$$ but only form $$$3$$$ unique strings.

Over all $$$26^{\#\text{ of ?'s}}$$$ ways Hina can fill in the question marks, what is the expected number of unique strings that can be formed?

Meruem the ant is on an infinite brick wall, where every brick is a $$$1$$$ by $$$2$$$ inch rectangle, and each row is a duplicate of the previous row, shifted $$$1$$$ inch to the right. Every day he starts on the lower left point of a brick and travels in a straight line to the point $$$x$$$ inches to the east and $$$y$$$ inches to the north. How many bricks does Meruem pass through?

There's an explorer entering a magic labyrinth. As he enters, the entrance closes behind him and the labyrinth starts filling up with poisonous gas. Once he realises, he asks for your help so that he is exposed to the minimum amount of gas possible.

The labyrinth can be represented as a simple directed graph of $$$n \le 100$$$ vertices, numbered from $$$1$$$ to $$$n$$$, and $$$m \le 1000$$$ edges. The explorer is initially at vertex $$$1$$$. The amount of gas in each vertex can be represented as an array $$$a$$$, being $$$a_{i}$$$ the amount of gas at vertex $$$i$$$.

The gas will last at the labyrinth for exactly $$$k$$$ seconds. After that, the explorer will be free again. Each second, the explorer can either move to any vertex $$$v$$$ such that there exists an edge from $$$u$$$ to $$$v$$$ or stay at vertex $$$u$$$, being $$$u$$$ the vertex he is currently at, but only after being exposed to the gas in vertex $$$u$$$. Due to the magic of the labyrinth, each second, after the explorer is exposed to the gas at vertex $$$u$$$, the array $$$a$$$ will cyclically shift once to the left (i.e. simultaneously set for all $$$i$$$, $$$1 \leq i \lt n$$$, $$$a_{i}$$$ to $$$a_{i+1}$$$ and $$$a_{n}$$$ to $$$a_{1}$$$).

You need to determine what is the minimum amount of gas the explorer can be exposed to. Note that the explorer doesn't need to return to vertex $$$1$$$.

After listening to Mask by Dream, Asahi decided to start wearing a mask and a green hoodie. Unfortunately, this did not change the fact that she is still failing all her classes. Thus, Asahi now turns to you for help on her test.

In the test, Asahi is given a tree with $$$N (1 \le N \le 5 \cdot 10^5)$$$ nodes. The $$$i$$$th edge in the tree between nodes $$$u_i$$$ and $$$v_i$$$ has a weight of $$$w_i (0 \le w_i \le 10^9)$$$. The goal is to choose two node disjoint (cannot share any nodes) paths in the tree. Asahi's score will then be $$$\text{min}(\text{sum of weight of path 1}, \text{sum of weight of path 2})$$$. Asahi wants to score as high as possible on the test, so she asks you to find her maximum possible score.

Note the higher time limit than usual.

You are given an array $$$a$$$ of $$$n$$$ integers and $$$q$$$ queries to process.

Each query consists of three integers, $$$l$$$, $$$r$$$ and $$$x$$$, and indicates that each value of $$$a$$$ between $$$l$$$ and $$$r$$$, inclusive, is incremented by $$$x$$$. Note that the updates are not independent, so the increments persist for the following queries.

After each query, print the minimum $$$i$$$ that satisfies the following conditions:

• $$$2 \leq i \leq n$$$.
• $$$a_{1} + a_{2} + \ldots + a_{i - 1} \leq a_{i}$$$.

Touya Mochizuki was struck by a bolt of lightning. Fortunately, instead of dying, he is sent to another world with his smartphone. Unfortunately, even in this generic new fantasy world, Touya is still unable to escape from generic range query problems!

Touya has an array $$$a$$$ of length of $$$N$$$ ($$$1 \le N \le 2 \cdot 10^5$$$). He is now tasked with $$$Q$$$ ($$$1 \le Q \le 2 \cdot 10^5$$$) queries of two possible types. Queries of type $$$2$$$ will add a value $$$v$$$ to every $$$a_i$$$ for $$$l \le i \le r$$$. Queries of type $$$1$$$ will ask him what is the sum of all subarrays in the range $$$l \dots r$$$, that is to say, $$$\displaystyle \sum_{i=l}^{r} \sum_{j=i}^{r} \sum_{k=i}^{j} a_k$$$. Since these values can be quite large, please output them modulo $$$10^9 + 7$$$

You are given $$$n$$$ numbers $$$a_1, a_2, \dots, a_n$$$. How many numbers between $$$l$$$ and $$$r$$$ $$$(1\le l \le r\le10^{18})$$$ can be expressed in the form $$$\displaystyle\sum_{i=1}^{n}\lfloor a_ix\rfloor$$$, where $$$x$$$ is a real number, and $$$\lfloor y\rfloor$$$ denotes greatest integer less than or equal to $$$y$$$?

To celebrate the airing of Tonikawa Season 2, the Yuzaki couple has decided to host a giant game of bingo at the bathhouse. There are $$$N$$$ people, conveniently numbered with an ID from $$$1$$$ to $$$N$$$, each with a $$$5$$$ by $$$5$$$ bingo board where each number is in the range $$$[1, k]$$$. Nasa will then call out the numbers from $$$1$$$ to $$$k$$$ in a certain order. The first person to have a full row or column of numbers that have been called already on their board wins (for simplicity diagonals are ignored in this problem). In the case of a tie, the person with the lowest ID wins. Nasa is going to ask you many different orders, and for each order he wants you to find the person who will win.