Tyger has an array of positive integers $$$a_1$$$, $$$a_2$$$, ..., $$$a_n$$$ of length $$$n$$$, on which he wants to perform a series of operations. An operation on an array consists of removing a number $$$x$$$ from the array and adding two positive integers $$$y$$$ and $$$z$$$ to the array such that $$$x = y + z$$$ and the addition doesn't yield any carries. Find the maximum possible number of operations Tyger can perform on the array.

Tyger is practicing on an archery target, which is an $$$n \times n$$$ grid of unit squares on the coordinate plane, where $$$n$$$ is odd. The target is arranged into concentric square rings around a single center square.

Each unit square is labeled by a pair of integers $$$(x, y)$$$. The target's bottom-left square is labeled $$$(x_0, y_0)$$$, and the square $$$i$$$ columns right and $$$j$$$ rows up $$$(0 \leq i, j \leq n - 1)$$$ is labeled $$$(x_0 + i, y_0 + j)$$$.

The outermost ring is worth $$$1$$$ point, and the score increases by $$$1$$$ for each ring toward the center. The single center square is worth $$$\frac{n + 1}{2}$$$ points. The squares outside of the target are worth $$$0$$$ points.

Tyger wants you to answer $$$q$$$ queries:

• 1 x y: Given that an arrow lands at the square labeled $$$(x, y)$$$, output its point value.
• 2 p: Given that an arrow scored $$$p$$$ points, find $$$\sum (x + y)$$$ over all squares labeled $$$(x, y)$$$ that are worth $$$p$$$ points.

LexMACS needs your help! There have been rumors of an in-person LIT, causing a multitude of desks to be purchased. Their arrangement on an $$$n$$$ by $$$m$$$ tile floor, where each tile is either fully below a desk or empty, can be shown through a binary grid, where a $$$1$$$ denotes the presence of a desk and a $$$0$$$ denotes the absence of a desk. The floor has walls on its edges, meaning that both competitors and proctors cannot go off the floor.

However, preemptive measures against cheaters have been placed. The room's proctor will initially teleport to an empty tile, and move to other adjacent, empty tiles as much as they want. A proctor is able to check a competitor and their desk for cheating if they are able to stand directly adjacent to the desk.

Help the proctor determine if a certain location for initial teleportation is effective for stopping cheaters, that is, whether the proctor can check each desk!

Cody wants to shoot a missile to hit Tyger. The missile starts at $$$(0, 0)$$$, and follows a parabolic trajectory under gravity, modeled by the equation $$$y = ax^2 + bx$$$, where $$$a \lt 0$$$ and the trajectory passes through the enemy's position $$$(E_x, E_y)$$$.

Between them, there are $$$n$$$ vertical obstacles. The $$$i$$$th obstacle is at x coordinate $$$x_i$$$ and spans a closed interval $$$[y_i, z_i]$$$. In other words, the obstacle can be seen as a segment connecting $$$(x_i, y_i)$$$ and $$$(x_i, z_i)$$$.

The missile moves continuously along the curve, but it can't pass through any obstacle.

Your task is to find a set of coefficients $$$(a, b)$$$ such that the missile hits Tyger and avoids all obstacles. If multiple such trajectories exist, output the one where $$$a$$$ is maximized. It is guaranteed that a solution exists.

You are given a tree with $$$n$$$ vertices, labeled from $$$1$$$ to $$$n$$$. Each vertex $$$i$$$ has a hidden label $$$v_i$$$, and the array of hidden values $$$[v_1, v_2, \ldots, v_n]$$$ forms a permutation of integers from $$$1$$$ to $$$n$$$.

You are not given the hidden values directly. Instead, you are given the structure of the tree and the weights of its $$$n - 1$$$ edges. The weight of the edge connecting vertex $$$i$$$ and vertex $$$j$$$ is $$$v_i \oplus v_j$$$, where $$$\oplus$$$ denotes the bitwise XOR operation.

Your task is to reconstruct the hidden labels $$$v_1, v_2, \dots v_n$$$ with the given information.

A left circular shift of a sequence moves its first element to the end. For example, the sequence $$$[1, 2, 3, 4]$$$ becomes $$$[2, 3, 4, 1]$$$ after applying a circular shift.

We can construct a grid $$$g$$$ of size $$$n$$$ by $$$n$$$ as follows: The first row of $$$g$$$ is $$$[1, 2, ..., n]$$$. For $$$i = 2$$$ to $$$n$$$, row $$$i$$$ is obtained by applying a left circular shift to row $$$i - 1$$$.

For example, when $$$n = 4$$$ our grid is:

1 2 3 4
2 3 4 1
3 4 1 2
4 1 2 3

You need to process $$$q$$$ queries of the following four types:

R x k: Apply a left circular shift on row $$$x$$$ by $$$k$$$ positions if row $$$x$$$ is a circular shift of $$$[1, 2, ..., n]$$$.

C x k: Apply an upward circular shift on column $$$x$$$ by $$$k$$$ positions if column $$$x$$$ is a circular shift of $$$[1, 2, ..., n]$$$.

S x y: Swap rows $$$x$$$ and $$$y$$$

Q x y: Output the value currently in row $$$x$$$, column $$$y$$$

You are given a simple connected graph with $$$n$$$ vertices and $$$n$$$ edges. A graph is beautiful if all of its connected components are trees of size $$$4$$$. A graph is good if a set of edges can be deleted from the graph to make it beautiful. For each of the $$$n$$$ edges, output whether the graph formed without the edge is good.

In the example above, since the red edges can be deleted to make the resulting graph *beautiful*, the graph above is an example of a *good* graph.

You are given an array $$$a$$$ of $$$n$$$ integers and a fixed integer $$$k$$$. Process $$$q$$$ queries of two types:

1. A count query, where you are given $$$l$$$ and $$$r$$$ and count the number of pairs $$$(i, j)$$$ such that $$$l \leq i \lt j \leq r$$$ and $$$a_j - a_i = k$$$.

2. An update query, where you are given $$$p$$$ and $$$x$$$ and set $$$a_p = x$$$.

Alice is chasing Bob through a network of $$$n$$$ rooms. The network is organized as a rooted tree of rooms, numbered from $$$1$$$ to $$$n$$$ and rooted at room $$$1$$$.

At the start of the game, Alice only has control of room $$$A$$$, and Bob is located in room $$$B$$$. It is guaranteed that room $$$A$$$ is the parent of room $$$B$$$.

Each day consists of two sequential phases, in the following order:

1. Bob must choose to either:

• Teleport to an ancestor of his current room that is $$$i$$$ edges away, where $$$0 \leq i \leq K$$$. The destination can't be controlled by Alice, but Bob may pass through intermediate rooms along the path that are controlled by Alice
• Move to any child of his current room that is not controlled by Alice

2. Alice takes control of exactly one room adjacent to her currently controlled territory. If the newly controlled room is the one Bob is currently in, Bob is captured and the game ends immediately.

Bob wants to maximize the number of days he evades capture, while Alice wants to capture him as quickly as possible. Both players play optimally and have perfect information about the state of the network at all times. Your task is to find the number of days Bob would survive, including the day he is captured.

You are given three integers $$$n$$$, $$$m$$$, and $$$k$$$. Your goal is to construct a not necessarily connected simple undirected graph with $$$n$$$ nodes and $$$m$$$ edges (which can be colored red, green, and blue) such that the score is at least $$$k$$$.

A "tetree" is defined as a selection of three edges such that they form a connected graph and connects 4 nodes and the three edges have different colors.

A "rainbow triangle" is defined as a cycle of length $$$3$$$ where each color is used exactly once.

The score of a graph is determined by the formula $$$$$$T - PR$$$$$$

where $$$T$$$ is the number of tetrees, $$$R$$$ is the number of rainbow triangles, and $$$P$$$ is an arbitrary number given in the input.