You are standing on a point with integer coordinates $$$(x_s, y_s)$$$. You want to walk to the point with integer coordinates $$$(x_t, y_t)$$$. To do this, you can walk along a sequence of line segments. But there is a swimming pool in your way. The swimming pool is an axis aligned rectangle whose lower left corner is on the point $$$(x_l, y_l)$$$ and the upper right corner is on the point $$$(x_r, y_r)$$$. You cannot ever cross the swimming pool, not even on the border. However, it is dark and you do not know the coordinates $$$(x_l, y_l)$$$ and $$$(x_r, y_r)$$$. Instead, you threw a rock into the pool which revealed that the point $$$(x_p, y_p)$$$ is in the pool (or on the boundary).

Find a way to walk from the start to the end point along a sequence of line segments, so that you never cross the swimming pool.

After lengthy meetings on the subject, the NCPC jury has decided that getting the contestants' blood pumping would result in better contests. Thus they figured it would be a good idea to compete for bragging rights in some kind of sport prior to the programming contest. For this they need some place to compete and to pick a sport to compete in. One of those is solved easily by drawing at random from a hat, which results in the chosen sport being baseball. That simply leaves the matter of making a suitable court to play on. The NCPC jury has access to a rectangular plot of land $$$a$$$ by $$$b$$$ meters in size. Furthermore, they have $$$N$$$ square tiles of grass $$$1$$$ by $$$1$$$ meter in size they can place on this plot to create the court. All grass tiles must be placed with their sides parallel to the edges of the plot.

The south-west corner of the land is chosen to be the batting point. For the placement of the grass tiles to constitute a valid court, two conditions must be met. Firstly, for any given tile the south and west sides must either lay directly against the edge of the plot or directly against another tile. This is required to make sure the ball doesn't exit and reenter the court while following a straight trajectory. Secondly, all tiles which have no adjacent tile to the north nor to the east must have the same manhattan distance from the batting point. This is to prevent batters from preferring some directions over others.

Your task is to find the number of ways to place the tiles so that it creates a valid baseball court.

Rhythm is an important part of music and it is crucial for aspiring musicians to gain understanding of it. As the skill of the musician advances, more complex rhythms are introduced to them. To ease the learning of musical passages, a method of simplifying rhythms can be helpful. One method is to reduce the rhythm into groups of twos and threes.

A rhythm is composed of multiple beats. A single beat does not make up a rhythm, as the beats depend on each other. The rhythm can be subdivided into smaller components. For example, a rhythm of $$$7$$$ beats can be subdivided into $$$4$$$ beats and $$$3$$$ beats, or alternatively into $$$2$$$, $$$3$$$, and $$$2$$$ beats. However, a rhythm of $$$7$$$ beats cannot be subdivided into $$$1$$$, $$$3$$$, and $$$3$$$ beats, since one of the components does not form a rhythm.

This leaves $$$2$$$ as the smallest group size of beats we can use to decompose a rhythm, but if we only use groups of size $$$2$$$ then we cannot have an odd number of beats. Adding $$$3$$$ as a group size allows us to decompose any rhythm, even if it has an odd number of beats.

Given the number of beats in a rhythm, provide one decomposition of the rhythm into groups of sizes $$$2$$$ and $$$3$$$.

You are playing a new card game. In the game you have two decks of cards each consisting of $$$N \cdot K$$$ cards labeled with an integer from $$$1$$$ to $$$N$$$, inclusive. Also, each type of card appears precisely $$$K$$$ times in each deck.

The rules of the game are simple. You shuffle both decks and place them face up in front of you, so at each point in time you see the top card in each deck. If the top cards are the same you can take them both and get one point. Otherwise you must discard either card. Your goal is to get as many points as possible.

You have just finished playing a round of this game and you want to know what the maximum score was, knowing the layout of both decks.

Carl the snake is in his burrow at the point $$$(0,0)$$$ in an infinite plane. He wants to visit the points $$$(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)$$$. These points must be visited in the order they are given, and Carl must end up at the point $$$(x_N, y_N)$$$. In one move, he can move one step up, down, left, or right. However, since he is a very long snake, he can never visit the same point more than once.

Your task is to find a sequence of moves such that Carl visits all the points in order, and never visits any point more than once. The points $$$(x_i, y_i)$$$ were generated uniformly at random.

The National Crop Protection Commission (NCPC) is dedicated to supporting local farmers by offering subsidies that are proportional to the area of their crop fields. Each farmer can have several crop fields, each uniquely shaped but geometrically defined as a polygon, bounded by fences that meet only at the corners.

In a classic display of political bureaucracy, and to incentivize well-shaped fields, the NCPC will subsidize each crop field based on the square of its area. This, to reward well-encapsulated fields. They now require a tool that calculates the sum of the squared areas of these polygons to ensure that the subsidies are distributed fairly.

Ingfríður is testing her new pet project website, Passwordle. The rules should be rather familiar for those who have played plenty of wordle, but let us review them here. The website picks a secret password that the user then has to guess. The password will be $$$N$$$ characters in length and the user will guess until they get it right, getting a higher score for fewer guesses. Each guess must be a string of $$$N$$$ characters. For each character in the guess, it will be coloured one of three colours. If that character matches the password character in the same position, the colour is green. If the character does not match, but that character is in the password somewhere else, the colour is yellow. Otherwise it is gray.

Ingfríður is of course very good at her own game, so she will always guess a password that could be the hidden password. That is to say her guess will always match the previous clues. Furthermore she knows her program never generates passwords with repeated characters, since that's insecure, and will incorporate this knowledge into her guesses.

You now receive some screenshots from her testing progress, but they got so terribly compressed you can only make out the colours and not the text itself. Furthermore it seems that she didn't manage to make any progress at all. She didn't get a single green square in the entire game and never found any more characters than she did in her very first guess, and quit out of frustration. So the number of yellow squares is the same on each row. Given this info, can you reconstruct a sequence of guesses she could have made? Or is it impossible and her program must be wrong? You may assume that Ingfríður never makes any mistake when playing, only when programming.

In an earlier contest, contestants had to solve a simple problem. They were given a string and had to print each unique substring of it, along with the number of occurrences it had in the original string. For example AB would print A 1 B 1 AB 1 and AAA would print A 3 AA 2 AAA 1.

When copying over this problem for reuse in this contest, several mistakes were made. The input constraints were changed significantly, making the problem absolutely impossible! Luckily this was partially cancelled out by the output validator being mangled as well. Now instead of checking for absolute correctness it only requires the number of occurrences of each character in the output to be correct. With a quick hotfix of applying run length encoding to the output, the problem was finally solvable again and the contest could continue on as planned. Right?

Svalur Handsome has finally graduated with a degree in computer science, and it couldn't have happened sooner. He has some rather unwise spending habits which he hopes will be more sustainable now that he can get a high paying job as a programmer. He has applied to a few places, and now has a contract in his hands that he could sign and start working almost immediately. But before he takes the offer he wants to figure out how long it could support his spending habits.

At the start of every day Svalur spends half of his remaining money, rounded up. The new job would pay $$$s$$$ ISK at the end of every $$$d$$$-th day, starting with the $$$d$$$-th day. He currently has $$$m$$$ ISK to spend as well.

Denise is designing a rainforest themed board game. The goal of the game is for each player to form a team of two characters and complete various challenges.

There are $$$N$$$ different characters numbered from $$$1$$$ to $$$N$$$. Each character $$$i$$$ has two attributes $$$p_i$$$ and $$$s_i$$$ (problem solving skill and strength). The numbers $$$p_i$$$ and $$$s_i$$$ are positive integers satisfying $$$1 \leq p_i, s_i \leq N$$$. Before the game starts, each player will pick two characters $$$i$$$ and $$$j$$$ to form a team. It is possible to pick two copies of the same character. The total problem solving skill and strength of the team will be $$$p_i + p_j$$$ and $$$s_i + s_j$$$ respectively.

In the game there are also $$$N$$$ challenge cards numbered from $$$1$$$ to $$$N$$$. Each of these also has two attributes $$$P_k$$$ and $$$S_k$$$. Denise has already designed the challenge cards and decided on the values of all numbers $$$P_1, P_2, \dots, P_N$$$ and $$$S_1, S_2, \dots, S_N$$$. However, the rules of the game assume that it is not possible for a player to form a team whose problem solving skill and strength are both the same as one of the challenge cards. In other words, the situation

$$$$$$p_i+p_j = P_k \text{ and } s_i+s_j = S_k$$$$$$

should never occur for any triple $$$i,j,k$$$ (note that $$$i$$$ can be equal to $$$j$$$).

The only thing left to do is to decide the $$$N$$$ distinct pairs $$$(p_1, s_1), (p_2, s_2) \dots, (p_N, s_N)$$$ such that $$$1 \leq p_i, s_i \leq N$$$ and the situation above never happens.

Jörmunrekur had found himself with some extra time on his hands, so he decided to try to find a new hobby. After discussing this with some of his relatives, his grandparents lent him a book with knitting guides and knitting patterns.

He wants to start with something big, so he decides to make a sweater. He has also picked out a pattern from the book that he will repeat around the circumference of the sweater. He wants the pattern to be centered and then repeat out towards the back in either direction, but never wants to have less than the full pattern on the sweater. He will not place any patterns that leaves the placements of patterns asymmetric. Now he has to know how much empty space he should leave at the back of the sweater to achieve this.

The empty space that is not covered by the patterns must be a contiguous (possibly empty) section at the back of the sweater.