Two frogs, Thomas and Waymo, are attempting to cross a pond.

Currently, they are both at the starting shore of the pond. In the pond, there are $$$N$$$ rows of unoccupied lily pads. In each row, there are two lily pads, a left lily pad and a right lily pad. One of them is afloat, and one of them is underwater. Also, each lily pad can hold at most one frog.

A frog may jump from the starting shore to an unoccupied and afloat lily pad in the first row, or from a lily pad in one row to an unoccupied and afloat lily pad in the next row, or from a lily pad in the last row to the ending shore.

When a frog jumps from a lily pad, that lily pad goes underwater, and the other lily pad in its row goes afloat. If a frog jumps from the right lily pad in a row, the right lily pad will go underwater, and the left lily pad will go afloat, and vice versa. Also, the frogs are polite. Thus, when one frog is mid-jump, the other frog will not jump.

Thomas and Waymo will jump in any order, obeying the above rules, until they both have made it to the ending shore. When a frog lands on a lily pad on the right, it gains a point. What is the maximum number of points total that Waymo and Thomas can gain?

You are a GothamChess subscriber. Playing a game of chess, you are nearing the end. It is your turn; you have a king and a rook, and your opponent has a king.

As every GothamChess subscriber knows, in this position, you need to sacrifice THE ROOOOOOOK!!!!!!!!

To sacrifice the rook, you must make a move such that the opponent's king can legally capture the rook on the next move. (Thus accepting the sacrifice)

Can you sacrifice the rook in one move?

Rules of Chess:

A chessboard is an $$$8$$$ by $$$8$$$ grid of squares. The columns are files, labeled $$$a$$$ – $$$h$$$. The rows are ranks, labeled $$$1$$$ – $$$8$$$. Kings and rooks are referred to as pieces, and they each occupy a square on the board. The location of a piece can be specified by the file, then the rank.

Each piece is controlled by one of the players. Specifically, in our problem, you control a king and a rook, and the opponent controls a king. On a player's turn, they select one of their pieces and move it.

A king can move one square horizontally, vertically, or diagonally. If it lands on an opponent's piece, it can capture it, but it may not land on an allied piece. Also, it may not move into check. (That is, it cannot move into a square where it could be captured on the opponent's turn.)

A rook can move any number of squares horizontally or vertically, but cannot move through other pieces. If it lands on an opponent's piece, it can capture it. It may not land on an allied piece.

Additional Constraints:

It is guaranteed that the positions are valid. The opponent's king is not under attack. That is, you will not be able to capture your opponent's king on your turn.

Castling is not possible in this position (understanding castling is unnecessary for solving this problem).

Unlike BAPC, Teamscode has enough funds to fill not just the gym, but the entire world with argon.

The world is represented by a grid of size $$$N \times M$$$. To help with filling the world with argon, you are given a 2D map, where $$$a_{i,j}$$$ represents the height of each of the cell in the $$$i$$$-th row and $$$j$$$-th column. All heights are distinct.

When you pour argon on a cell, it becomes filled with argon. Argon then flows from a filled cell to any of its adjacent cells (up, down, left, right) that have a lower height, causing them to fill with argon too. This filling process continues recursively until no further cells can be filled from that drop.

Determine the minimum number of cells you need to pour argon on to fill the entire world with argon.

You are given an undirected graph with $$$N$$$ nodes and $$$M$$$ edges. Each node is assigned a bitmask $$$A$$$ (where $$$1 \leq A \leq 2^{30} - 1$$$). A node is said to 'contain' color $$$c$$$ (with $$$0 \leq c \leq 29$$$) if the bit in the $$$2^c$$$ place in the binary representation of $$$A$$$ is $$$1$$$. This expression using bitwise operators would evaluate to true if and only if $$$A$$$ contains the color $$$c$$$.

$$$$$$ (A \ \& \ (1 \ll c)) == (1 \ll c) $$$$$$

A path between two nodes $$$i$$$ and $$$j$$$ is considered valid if there exists at least one color $$$c$$$ such that every node along the path (including $$$i$$$ and $$$j$$$) contains color $$$c$$$.

Your task is to determine, for every node $$$i$$$, which nodes $$$j$$$ are reachable from $$$i$$$ via some valid path. Output an $$$N \times N$$$ matrix of characters, where the $$$j$$$th character in the $$$i$$$th row is '1' if there exists at least one valid path from node $$$i$$$ to node $$$j$$$, and '0' otherwise.

You are in the Squid Game! As the music turns on, you hear a command from the guards.

"The game you will be playing is Mingle. You must form groups of 2 and go into the rooms."

The room has $$$n$$$ players and $$$m$$$ pairs of friends. Two players $$$u$$$ and $$$v$$$ can team up if either of the following conditions are met:

• $$$u$$$ and $$$v$$$ are friends
• $$$u$$$ and $$$v$$$ share a mutual friend

Time is ticking! Being the hero, your goal is to pair as many people up as possible.

To pass the time, Bessie the cow and her friend Elsie like to play a version of a game they saw at the county fair.

To start, Bessie puts three inverted shells on a table and places a small round pebble under one of them. Bessie then proceeds to swap pairs of shells, while Elsie tries to guess the location of the pebble.

But Elsie has gotten too good at this game. So Bessie summons a floating grid of $$$2^n \times 2^n$$$ keys. The rows are labeled $$$0 \dots 2^n - 1$$$, and the columns are labeled $$$0 \dots 2^n - 1$$$.

Bessie then points to $$$m$$$ of the keys and tells Elsie to keep track of the keys.

Afterwards, Bessie uses her super cow powers to perform $$$k$$$ operations. In each operation, she selects $$$x, y$$$. Then, the grid of keys is partitioned into subgrids with $$$2^x$$$ rows and $$$2^y$$$ columns. Then, she rotates each subgrid $$$180$$$ degrees and joins the subgrids back together. After performing all $$$k$$$ operations, she asks Elsie to guess the locations of the keys she pointed to at the beginning.

Elsie could not keep up with Bessie's super cow powers. Help Elsie determine the locations of the keys that Bessie pointed to.

Consider a grid $$$c$$$ with $$$n$$$ rows and $$$m$$$ columns. The top-left corner of $$$c$$$ has coordinates $$$(0, 0)$$$, and the bottom-right corner has coordinates $$$(n-1, m-1)$$$.

The cell with coordinates $$$(i, j)$$$ of $$$c$$$ has cost $$$c_{i, j}$$$.

From $$$c$$$, create a big grid $$$C$$$ with $$$N\cdot n$$$ rows and $$$M\cdot m$$$ columns. The top-left corner of $$$C$$$ has coordinates $$$(0, 0)$$$, and the bottom-right corner has coordinates $$$(N\cdot n-1, M\cdot m-1)$$$.

The cell with coordinates $$$(i, j)$$$ of $$$C$$$ has cost $$$c_{\left \lfloor \frac{i}{N} \right \rfloor, \left \lfloor \frac{j}{M} \right\rfloor}$$$. $$$^{\text{∗}}$$$

A path through $$$C$$$ starts at $$$(0, 0)$$$, ends at $$$(Nn-1, Mm-1)$$$, and moves only right or down. The cost of the path is the sum of the costs of the cells visited.

What is the minimum cost of a path through the big grid $$$C$$$?

Bocchi has recently learned about the Fibonacci sequence (or as she likes to call it, Fibocchi sequence), and has now proclaimed herself the master of the Fibocchi sequence!

Ryou has decided to test this by giving Bocchi the $$$2$$$ starting numbers, $$$a$$$ and $$$b$$$, of the sequence $$$f_k$$$ and asking her to generate $$$n$$$ terms of the sequence. More formally, Ryou wants Bocchi to generate the sequence $$$f_k$$$ defined by the recurrence relation $$$$$$f_1 = a; f_2 = b; f_k = f_{k - 1} + f_{k - 2} \, (2 \lt k \leq n).$$$$$$

But to really test if Bocchi is a Fibocchi master, she has added a twist! She wants Bocchi to answer $$$q$$$ queries regarding the sequence after performing some number of operations on it. An operation is defined by adding $$$1$$$ to $$$f_k$$$ for some $$$k$$$ $$$(1\leq k\leq n)$$$, then recalculating the recurrence starting from index $$$k + 1$$$.

For example, if $$$f = [2, 1, 3, 4, 7]$$$, Bocchi can apply an operation with $$$k = 3$$$, turning $$$f$$$ into $$$[2, 1, 4, 5, 9]$$$.

Note that if $$$k = 1$$$ is chosen, $$$f_2$$$ does not change.

Ryou will give Bocchi $$$q$$$ queries of the following form:

Given $$$a$$$ and $$$b$$$, find the minimum number of operations needed to make $$$\sum_{k = 1}^n f_k$$$ equal to $$$x$$$ or $$$-1$$$ if this is not possible.

Bocchi finds these advanced operations too difficult. Please help her prove her title as the Fibocchi master!

You have been hired as a Temporary Part-Time Assistant Manager of the Executive Municipal Internet Service Provider Administration!

Your job is to set up cell towers to provide internet to homes in The City.

The City can be visualized as a very large rectangular grid. The rows are numbered starting from $$$1$$$, from south to north. The columns are numbered starting from $$$1$$$, from west to east. There are also $$$n$$$ houses on this grid, denoted by $$$(r, c)$$$, where $$$r$$$ is the number of the row the house is in, and $$$c$$$ is the number of the column the house is in.

You are trying to install cell towers for the houses. However, the cell towers have a peculiar set of rules.

A cell tower can only provide signal to a right isosceles triangle that extends south and east from it. More formally, a cell tower at $$$(r, c)$$$ covers all houses $$$(r', c')$$$, such that $$$c \leq c'$$$, and $$$r' + c' \leq r + c$$$.

If a cell tower is set at $$$(3, 1)$$$, it can only provide signal to houses at $$$(3, 1), (2, 1), (1, 1), (2, 2), (1, 2), (1, 3)$$$. Here is a diagram of the range of a cell tower at $$$(3, 1)$$$:

Because the terrain gets rougher the further north you go, the cost of a cell tower is equal to the number of the row it is installed. For instance, installing a cell tower at $$$(3, 1)$$$ has a cost of $$$3$$$.

A cell tower may be installed on top of houses, and it will still provide coverage as usual (despite protests from homeowners).

The City is a bit indecisive about how many cell towers they would like to install. Thus, for every $$$i$$$ from $$$1$$$ to $$$n$$$, find the minimum cost to cover all houses with at most $$$i$$$ cell towers.

In the following undirected graph, count the number of walks from $$$A$$$ to $$$A$$$ that visit nodes $$$A$$$, $$$B$$$, $$$C$$$, $$$D$$$ exactly $$$a$$$, $$$b$$$, $$$c$$$, $$$d$$$ times, respectively, modulo $$$998244353$$$.

You encounter a triangle. The first thing you can tell about it is that it is most certainly not Japanese.

The triangle contains $$$n$$$ rows, the $$$i$$$th from top contains $$$i$$$ numbers, $$$a_{i,1}, a_{i,2} \dots a_{i,i}$$$. The numbers are not written in Japanese.

The triangle then speaks to you, but not in Japanese. It wagers $$$100$$$ points that you cannot find the sum of the numbers in each of its rows.

You confidently accept the challenge. But before you begin, the triangle puts on a not Japanese shirt, obscuring all rows from $$$1$$$ to $$$n-1$$$, and only the numbers in row $$$n$$$ remain visible.

Fortunately, the shirt has some not Japanese text that reveals an important property: Every hidden number is the minimum of two numbers directly below it. In other words, $$$a_{i,j} = min(a_{i+1,j}, a_{i+1, j+1})$$$ for all $$$1 \leq i \leq n-1$$$ and $$$1 \leq j \leq i$$$.

Can you find the sum of the not Japanese numbers in each not Japanese row of the not Japanese triangle?

Yam is racing robots! Unfortunately, his robots are out of control, so he has to stop them before they crash.

Yam has $$$N$$$ robots, each with speed $$$a_i$$$ ($$$a_i$$$ is non-decreasing). They are all positioned at the start (i.e., position $$$0$$$) of a track of length $$$L$$$ meters. For each subarray, Yam does the following:

First, he divides the robots in the subarray into $$$k$$$ groups of robots, where each robot must belong to exactly one group, and each group has at least one robot. Every second, every robot will advance $$$a_i$$$ meters forward. Then, after every second, Yam chooses a group and shoots every robot in the group, stopping them from moving forward in the future. If any robot crosses the end of the track (i.e., its position is strictly greater than $$$L$$$), then it spontaneously combusts. Define the score of this subarray as the maximum number of groups that Yam can form while ensuring that no robot spontaneously combusts. (Yam can choose to group the robots strategically to maximize the score.)

Yam thinks this is too easy, so he challenges you to find the sum of scores over all subarrays.