We have $$$n$$$ squares, $$$s_1, s_2, \dots, s_n$$$, on a 2D plane. Now, we want to draw a square $$$S$$$ such that for each of the existing squares, the intersection with $$$S$$$ is also a square with nonzero area.

Formally, we want to find a new square $$$S$$$ such that for each $$$i \in \{1, 2, \dots, n\}$$$, the intersection $$$S \cap s_i$$$ is also a square with nonzero area.

Print out any valid square $$$S$$$.

Unable to avoid the AI takeover, Bocchi has recently started studying neural networks! She has learned a special type of neural network that can be modeled as a specific type of graph.

The structure of this graph is special: it consists of $$$n + 2$$$ nodes, each with some value $$$v_i$$$. Bocchi also notices that all values from $$$0$$$ to $$$n + 1$$$ are present. The graph has a starting node, $$$d$$$ layers of nodes, and an ending node*. Nodes within each layer aren't connected, but each node in layer $$$i$$$ is connected to every node in layer $$$i + 1$$$ with a directed edge, with the starting node being connected to every node in layer $$$1$$$ and every node in layer $$$d$$$ being connected to the ending node. The starting node's value is always $$$0$$$, while the ending node's value is always $$$n + 1.$$$

After studying the neural network for some time, she deems the efficiency of a certain path from the starting node to the ending node to be the $$$\text{MEX}$$$** of the values on the path.

For example, suppose that there are two layers, the first consisting of nodes of values 4 and 1, and the second consisting of nodes of values 2 and 3. The starting node is 0 and the ending node is 5.

Perusing all of the options:

• The path $$$0\rightarrow4\rightarrow2\rightarrow5$$$ has an efficiency of $$$1$$$.
• The path $$$0\rightarrow4\rightarrow3\rightarrow5$$$ has a efficiency of $$$1$$$.
• The path $$$0\rightarrow1\rightarrow2\rightarrow5$$$ has a efficiency of $$$3$$$.
• The path $$$0\rightarrow1\rightarrow3\rightarrow5$$$ has a efficiency of $$$2$$$.

To see how well her neural network works, she gives her neural network an evaluation score. Specifically, the evaluation score is equal to the maximum efficiency over all paths from starting node to ending node. In the example neural network above, the path with the highest efficiency is the path $$$0\rightarrow1\rightarrow2\rightarrow5,$$$ so the evaluation score of the neural network is $$$3.$$$

After spending all day building her neural network, she is too tired to evaluate it. She now gives you her neural network and asks you to find the evaluation score of it!

Notes:

* For this problem, the starting and ending nodes do not count as layers.

** The $$$\text{MEX}$$$, or minimal excluded, of a sequence is the smallest non-negative integer not present in that sequence.

You are given a tree with $$$n$$$ nodes, rooted at node $$$1$$$. The $$$i$$$-th node has a strength of $$$w_i$$$, meaning it can withstand a total of $$$w_i$$$ kgs of snow in its subtree. If a node has more than $$$w_i$$$ kgs of snow in its subtree, it is too heavy, and it will break.

At the end of every day, snow falls on every node. That is, $$$a_i$$$ kgs of snow fall on the $$$i$$$-th node. The snow falls instantaneously.

You may reinforce up to $$$k$$$ nodes, making them unbreakable. You wish to strengthen the tree such that the maximum possible amount of snow will gather on the tree before it breaks. The tree breaks once any of its nodes break. Snow that falls on the day that it breaks does not count.

For each $$$k = 0, 1, \dots, n-1,$$$ output the maximum weight of snow that the tree can gather.

Alice and Bob are playing a game on an array $$$A$$$ consisting of $$$n$$$ positive integers. Alice picks some $$$i$$$ and sets $$$A[i]$$$ to $$$A[i]$$$ $$$|$$$ $$$x$$$. Then, Bob picks some $$$j$$$ and sets $$$A[j]$$$ to $$$A[j]$$$ $$$\&$$$ $$$y$$$. Note that each player only plays one turn. The score of the game is equal to the sum of $$$A$$$.

Alice wants to minimize the score while Bob wants to maximize the score. Find the score of the game if both players play optimally.

Farmer John considers himself a bit of a philosopher! He often spends time daydreaming about his favorite thought experiments, including the trolley problem.

One day, Farmer John needed to deploy $$$n$$$ trolleys to transport haybales, where the trolleys are listed in ascending order of weight from left to right, and the trolleys move to the right. Unfortunately, Farmer John's prized cows were taking an afternoon nap and ended up right in the way of the trolleys!

There are two tracks in front of Farmer John's array of trolleys, with each track having $$$a$$$ and $$$b$$$ cows of their own respective weights. Farmer John can choose to redirect trolleys to either track at will. If a trolley is heavier than the cow it collides with, the cow will wake up and run off the track. Otherwise, the cow will remain sleeping on the track. The trolley will always derail after a collision.

Farmer John has no time to deliberate the ethics of redirecting the trolleys; he just wants to prevent as many cows as possible from being awoken. Can you help him out?

You are given $$$t$$$ test cases. Each test case is a sequence of five non-negative integers. Your goal is to reduce all five numbers to zero by applying a sequence of operations at minimum total cost.

There are two types of operations:

• Operation $$$1$$$: Select any three consecutive elements and decrease each by $$$1$$$. Cost: $$$1$$$.
• Operation $$$2$$$: Select any four consecutive elements and decrease each by $$$1$$$. Cost: $$$2$$$.

For each test case, determine the minimum total cost required to reduce all five integers to zero. If it is impossible, output $$$-1$$$. 

Determine the maximum possible width of any possible binary tree constructed with $$$n$$$ nodes.

- A binary tree is defined as any tree where all nodes have at most 2 children.

- The width of a tree is defined as the maximum number of nodes at any level of the tree. Null nodes do not contribute to the width of the tree.

You are given an array $$$a$$$ of $$$n$$$ integers $$$a_1,a_2,a_3,\ldots,a_n$$$.

Define $$$f(k)$$$ as the $$$\text{MEX}$$$ of the maximum value of all subarrays of length $$$k$$$.

Print $$$f(k)$$$ for each $$$k$$$ from 1 to n.

More formally, output the values of $$$\text{MEX}^{n-i}_{j=0} \max^{i}_{k=1} a_{j+k}$$$ for all $$$i$$$ from $$$1$$$ to $$$n$$$.

A subarray of length $$$k$$$ is an array that contains exactly $$$k$$$ elements obtained by removing a prefix and a suffix from the original array.

The $$$\text{MEX}$$$, or minimal excluded, of a sequence is the smallest non-negative integer not present in that sequence.

Mr. G's students are hanging up pennants! There is a wall with $$$L$$$ slots, and has $$$N$$$ pennants hung on slots $$$1$$$ through $$$N$$$. The pennant in the $$$i$$$-th slot has rank $$$r_i$$$ (not necessarily distinct), and the pennants are ordered by rank, so $$$r_i\le r_{i+1}$$$ for $$$1\le i \lt N$$$. In one second, the students can either take a pennant off of a slot, or put a pennant of any rank onto the wall in any slot.

The students have put up all the pennants already, if they find $$$a$$$ extra pennants of rank $$$k$$$, then they must rearrange the pennants such that every pennant is on the wall, and the pennants are ordered by rank. It is guaranteed that this is always possible in finite time.

Mr. G has $$$q$$$ queries of $$$(a,k)$$$, and wants to find out the minimum time the students will take for each combination of $$$(a,k)$$$. Note that the pennants are not actually rearranged, that is, the queries are independent of each other.

note: the above image is not intended to be an accurate representation of the problem.

Alter's friends are playing a game of castlefall! The game has $$$N$$$ players, labelled $$$1$$$ through $$$N$$$, and $$$M$$$ words, labeled $$$1$$$ through $$$M$$$. Each player is assigned to exactly one of two non-empty teams of possibly different sizes, and each team is given one of the $$$M$$$ words. Note that the two teams are given different words.

Since players do not know which team they are on, each player has announced a clue. Specifically, the $$$i$$$-th player has a word in the set $$$S_i$$$.

Since Alter doesn't want to think, he asks you to help him answer the following question: Is it possible to uniquely determine the teams and the words corresponding to each team? If so, Alter will also ask you what the words are.

You are given a connected weighted graph with $$$n$$$ vertices and $$$m$$$ bidirectional edges. You are to traverse from vertex $$$s$$$ to vertex $$$t$$$.

Across all (not necessarily simple) paths from $$$s$$$ to $$$t$$$, count the number of distinct sums of weights along those paths modulo $$$k$$$.

Given a 2D grid with obstacles, you want to move a robot to point $$$B$$$. To do this, you can summon robots at spawn points marked as $$$A$$$. (As long as it isn't obstructed)

You may also instruct robots to move. A robot can move into an adjacent cell if there is empty space in that cell. However, if there is an obstacle, the movement will be unsuccessful. If there is a robot, the robot is able to push it, as long as the cell it is pushed to is empty. If there is another robot or an obstacle, the movement is unsuccessful. The outside of the grid is made of obstacles, so robots may not move outside the grid.

After a robot is instructed to move, even if the move was unsuccessful, the robot will self destruct and cease to exist, leaving an unoccupied cell. A pushed robot does not self destruct.

Here are $$$5$$$ examples of what happens when the red robot moves to the right:

Your goal is to find the minimum number of robots you need to summon in order to move a robot to point $$$B$$$ (Moving a robot to point $$$B$$$ only for it to immediately self destruct does not count).

If it is impossible to move a robot to point $$$B$$$, print $$$-1$$$.

Additionally, it is guaranteed by the test data that if it is possible to move a robot to point $$$B$$$, it can be done so in at most $$$4 \cdot 10^{18}$$$ summoned robots.