Maya wants to organize the training data for her machine-learning model. There are $$$N$$$ files that she can use, and each file has a size of $$$s_i$$$ bits. Her computer has a limited amount of storage, so she can only allocate $$$X$$$ bits toward storing all of the training data.

What is the maximum number of files that she can use for her model?

Tanner is speedrunning the hit video game The Legend of Zelda: Tears of the Kingdom, where his goal is to beat the game as fast as possible. To keep track of his progress, he creates splits for each important objective in the run.

Tanner wants to know his improvement between runs for certain splits. Given $$$N$$$ different runs composed of $$$K$$$ splits, with $$$t_i$$$ being the time in seconds for each split, for each of $$$Q$$$ different queries for a certain split $$$s$$$, please output the maximum improvement between any two times for a given split.

You are given the three-dimensional Cartesian coordinates $$$(x_i,y_i,z_i)$$$ of $$$N$$$ robots. Find the minimum area difference between two distinct triangles that can be formed by taking three of these robots as vertices.

The time given for this problem is double the usual amount (2s).

Kafka wants to cause some trouble at the Xianzhou Luofu. She is currently at the Ambrosial Arbor, which is a large tree with $$$n$$$ ($$$2 \le n \le 10^5$$$). In the tree, node $$$i$$$ has an explosion value of $$$a_i$$$ ($$$1 \le a_i \le 10^8$$$), and the edges all have weight $$$1$$$. If a simple cycle is created in the tree, it will explode with a magnitude of the sum of edge weights on the cycle. In order to carry out her terrorism, Kafka will add an edge between two nodes $$$u$$$ and $$$v$$$ with weight $$$a_u + a_v$$$, thus creating a simple cycle and exploding the tree. Note that $$$u \ne v$$$.

As a wanted terrorist, Kafka wants to cause as large an explosion as possible, so please tell her the largest magnitude explosion that she could cause by adding any edge.

Rowlet recently built an AI duck! It can't do much besides walk right now, but maybe one day its full capabilities will be realized. For now, however, it is able to walk the shortest path between any two points on a grid. To make sure the duck can walk properly, Rowlet wants it to go from point $$$(1, 1)$$$ to $$$(N, M)$$$. Now, Rowlet is interested in how many possible paths the duck might take. But the owl realized the number might be too large, so the duck was given $$$K$$$ squares it must pass through. Satisfied, Rowlet took a break and went to get some more fruit (because who doesn't love fruit).

Unfortunately, two problems arose! First, Rowlet's $$$K$$$ coordinates were corrupted, meaning that it may no longer be possible to go from $$$(1, 1)$$$ to $$$(N, M)$$$ and go through the $$$K$$$ extra squares without taking steps "backwaords" (to the left or downwards). Second, Rowlet realized that the number of possible paths is still too large. Since Rowlet is still on a fruit break (who knows how long owls take to eat fruit), you are asked to find the number of suitable paths modulo* $$$998244353$$$.

*$$$a \mod b$$$ is defined as the remainder left over when dividing $$$a$$$ by $$$b$$$. For example,

$$$2 \mod 4 = 2$$$ becuase $$$2=0*4+2$$$

$$$5 \mod 4 = 1$$$ because $$$5=1*4+1$$$

$$$10^9 \mod 998244353 = 1755647$$$ because $$$10^9=1*998244353+1755647$$$, so if the answer is $$$10^9$$$, print $$$1755647$$$ instead.

Silly Nilly the intelligent robot has just bought a new shelf with $$$L$$$ levels and wants to fill it with stuffed animals. She currently has $$$P$$$ piles of $$$S_i$$$ stuffed animals each, but since she is a perfectionist, Silly Nilly insists that each layer of the shelf must contain the same number of stuffed animals.

Since she has limited mobility as a robot, in one operation, Silly Nilly can either remove a stuffed animal from any pile or add a stuffed animal to any pile (she has more stuffed animals outside of these piles). Determine the minimum number of operations it will take for Silly Nilly to arrange her piles so that any $$$L$$$ piles will have the same number of stuffed animals in them.

Neo-Bot, an advanced pathfinder dog, is trapped in a 2D grid. He is at coordinates $$$(0, 0)$$$ and needs to get to $$$(x, y)$$$ to escape. There are also $$$M$$$ lines going from $$$(x1_i, y1_i)$$$ to $$$(x2_i, y2_i)$$$. These lines lie along the grid axes, so either both the x's or both the y's will be identical. Neo-Bot can only travel along the lines. Output how far Neo-Bot travels until he makes it to the end, or -1 if he never reaches there.

Sara has trained an AI model to generate pictures of kittens given certain verbal prompts. As a very basic form of evaluation for her model, she decides to take the absolute value of the difference $$$D$$$ between the sums of all pixels in a region of the image $$$A$$$ her model generates and the real-life reference photo $$$B$$$ that she chooses. $$$A$$$ and $$$B$$$ are both square images $$$N$$$ pixels long on each side.

Given $$$Q$$$ queries consisting of two pairs of coordinates $$$(r_1,c_1)$$$ and $$$(r_2,c_2)$$$, print out the value of $$$D$$$ for the rectangular regions in $$$A$$$ and $$$B$$$ bounded by $$$(r_1,c_1)$$$ and $$$(r_2,c_2)$$$ (including $$$(r_1,c_1)$$$ and $$$(r_2,c_2)$$$).

Thomas the Train, an artificial intelligence technology, is traveling between his $$$N$$$ train stations labeled $$$1$$$ to $$$N$$$, and he starts at station $$$1$$$. He knows that today, for each station $$$i$$$, one passenger will arrive at station $$$i$$$ at time $$$T_i$$$. It takes an amount of time $$$D_{i, j}$$$ to drive from station i to station j. Note that $$$D_{i, j}$$$ may differ from $$$D_{j, i}$$$.

Since Thomas is not very intelligent, Thomas travels to a station only if he will pick up a passenger there, and thus he does not take a shorter path to a station involving intermediate stations without picking up passengers there. Furthermore, the passengers are very impatient and unwilling to wait at a station for any amount of time, so Thomas must pick up passengers at the exact time at which they arrive. He can wait at a station for as long as he wants. Help Thomas determine the maximum number of passengers he can pick up today!

A word search is a puzzle in which the objective is to find certain words hidden within a rectangular grid of letters. Words must be in a straight line, meaning that they are horizontal, vertical, or diagonal.

A special word search is a word search that allows words to wrap around from one side of the word search to the opposite side. Like a regular word search, words must be in a straight line (horizontal, vertical, or diagonal). The same spot in the word search cannot be used in the same word twice.

For your computer science class final project, you decide to build and train an AI model to detect words in a special word search. However, you must label pieces of training data for the model. To do this, for every given $$$N$$$ x $$$N$$$ board of the special word search and an array of $$$K$$$ words, you must determine how many of those words appear in the word search.

Note that there may be words contained inside other words, such as "sea" and "seashell". There may also be multiple occurrences of the same word, but each word should only be counted once.