Oscar enjoys watching crime movies very much. He admires all those villains for how creative some of them can get. He would like to show off some of his creativity, too. Sadly, he's quite inexperienced and he is unable to come up with his own kind of a trick. So he would like to get inspired by some old trick. He always loved watching criminals cutting out the letters from newspapers and putting them together to form a ransom text. However, Oscar ain't a total rip off so he came up with his own variant of this trick. He feels that putting the text together letter by letter is simply too boring and time consuming. So he decided to create his ransom note by cutting out whole words.

Oscar bought several mainstream newspapers, therefore he has practically unlimited source of paper to cut the words from. He can cut out any particular word as many times as he wishes. However, he's still limited by the set of words which appear in the newspapers. The trouble is that some words are just not used in the newspapers at all. To make his job a bit easier, he decided to erase all punctuation and all whitespace characters from the ransom note and ignore the case of characters. He also allows the cut-out words to overlap, as far as the overlapping parts contain the same text. Oscar now wonders how many words at minimum he has to cut out from the newspapers to put together the ransom note.

In a very distant land called Midsommer, there is a cute little river delta. Deep purple acid flows in the river, so it is impossible to swim there. There are several islands in the delta and bridges connecting some pairs of them. Every bridge is assigned a danger level, which measures how dangerous it is to travel across that bridge, higher levels being more dangerous.

A detective, and as matter of coincidence also a mystery novels author, Richard Hradecki often needs to travel between the islands to pursue his detective cases. Among all possible paths, he prefers to choose the safest one, which means the highest danger level of a bridge on the path must be as low as possible.

For planning purposes, Richard usually asks you to find him the safest path from one island to another island where he is investigating. To be able to satisfy his requests, you have to continuously register three types of events:

• A new bridge between two islands was just built by members of a local tribe.
• A big pink fluffy acid bear Lug appears and destroys a bridge.
• Richard asks you to find the safest path between two islands.

Rabbits are small mammals in the family Leporidae of the order Lagomorpha. So says Wikipedia. Correctly. The corollary is that they are not boring, for they are all original and well organized. The rabbits on our farm live in guarded corrals whose borders are adorned with elaborate floral ornaments. Scores of adorable orange carrot clumps grow in the corrals. Rabbits reproduce quickly (it is a norm, hordes of rabbits are born each year) and our mentors are keen to locate them in the corrals effortlessly.

The corrals are well-ordered, they form one straight row of corrals. At the beginning of the first breeding season, some corrals may be unoccupied by rabbits. At the end of each breeding season, a well orchestrated relocation of rabbits is performed. The relocation is guided by a simple formula which depends on one positive integer parameter $$$K$$$, which may be chosen arbitrarily for each season. The relocation works on all corrals in parallel. In each corral, approximately one half of rabbits are removed from the corral and moved $$$K$$$ corrals down the row of the corrals. It does not matter whether the target corral is already occupied by some rabbits or not.

If a corral is too close to the end of the row (there are less than $$$K$$$ corrals down the row of corrals) then all the rabbits stay in the corral and are not moved anywhere.

Any corral can accommodate unlimited number of rabbits and there are always enough rabbits to breed successfully in any nonempty corral.

You are given a specification of which corrals are occupied and which are empty at the beginning of the first breeding season. Determine the minimum number of relocations needed to populate all corrals by rabbits.

You came to play with your dog to a water reservoir near your house. The dog likes to play with frisbee and since you are bored, you are curious how much time it takes your dog to catch the thrown frisbee and bring it back to you. So you hold your dog by your side and start your stopwatch. A few moments later, in time $$$T_f$$$ milliseconds, you throw the frisbee horizontally from height $$$H_f$$$ millimeters with velocity $$$V_f$$$ millimeters per millisecond. The vertical acceleration due to gravity specific to your frisbee is $$$1$$$ millimeter per millisecond squared.

In time $$$T_d$$$ milliseconds, you release the dog, whose maximal horizontal velocity is $$$V_d$$$. The dog races across ideally flat terrain and it is smart enough to minimize the time needed to bring the frisbee back to you. In order to achieve this, the dog can jump up to height Hd millimeters, and the jump does not influence its horizontal velocity. When the dog catches the frisbee, it immediately races back to you at full speed. You stop the stopwatch when the dog reaches you. Note that you stop the stopwatch even if the dog is in the air, as far as it is directly above the same place as in the beginning.

Your dog is special because it can gain horizontal velocity instantly (there is no speedup or slowdown) and it can change or reverse its horizontal velocity even during a jump while being airborne. Also, the vertical acceleration due to gravity specific to your dog is $$$3$$$ millimeters per millisecond squared.

For simplicity, assume that both the frisbee and the dog have negligible size. Also, we remind you that a general equation related to the movement of objects in this problem may be expressed in the form $$$s(t) = s_0 + v_0 t + \frac{1}{2}at^2$$$. Correct application of the ideas behind this equation is up to you, of course.

The huge trees in the Jumbarian jungle are strategically very important. The Jumbarian army headquarters have selected $$$N$$$ trees which are rather close to each other, and decided that a tree house will be built on each of these trees and it will be occupied by a single army unit. Movement of people and material between the tree tops will be supported by the system of bi-directional zip lines connecting pairs of tree tops. For safety reasons, no two zip lines can cross each other (when observed from a satellite).

$$$N$$$ units have already been chosen and a list of pairs of these units has been created. Units in a pair are expected to operate on the opposite ends of one zip line. The number of pairs in the list is one less than the number of units. It turns out the list ensures the connectedness of the area, i.e., after the units will be deployed to the trees, each unit will be able to reach any other unit using only zip lines that appear on the list.

All that remains to put this scheme to work is to install the zip lines between tree tops in such a way that will allow to deploy the units in accordance with the above rules.

In a royal casino, there is a huge room with many slot machines. The casino manager feels that this room may be too easy to navigate and so the people have a good chance of getting out of the room without losing much money. This needs to be fixed, therefore the manager designed a complicated system of slot machine positions and straight aisles between them, so people get lost there more likely.

The room's shape is roughly oval. If it was empty, the whole room could be observed from any inside point. Each of $$$N$$$ slot machines in the room can be represented as a point and it is assigned some unique profit value. The manager wants the slot machines to be connected by aisles in the way described below.

There already are at least $$$3$$$ machines built into the wall of the room. All these wall machines must be connected by straight aisles to create a closed perimeter path around the whole room. All of the remaining slot machines in the room (if there are any) will be inside the area enclosed by the perimeter path.

The room is to be divided by straight aisles into smaller separate areas, each enclosed by its own perimeter path. The division proceeds in two phases.

• The first phase is ruled by the following scheme. If there are at least $$$4$$$ slot machines on a perimeter path, two pivot machines have to be chosen. The first pivot machine is the most profitable one on the perimeter path. The second pivot machine is also on the perimeter path and it is the furthest one from the first one. The distance between machines is measured along the perimeter path and it is equal to the number of aisles one has to traverse to get from one machine to the other one. If there are two furthest machines, the more profitable one has to be selected as the second pivot machine. The area inside the perimeter path is divided into two areas by a new aisle which connects the pivot machines. The perimeter path of each of the two new smaller areas is the minimal part of the original perimeter path which together with the new aisle form a closed path encircling the smaller area. The scheme is applied repeatedly until no new area can be created.
• When the previous division phase is completed, the second division phase begins. It is ruled by the following scheme. For any area $$$A$$$ with no aisle inside it, consider all slot machines inside that area. If there are any, connect the most profitable one of them by aisles to all slot machines on the perimeter path $$$P_A$$$ of the area $$$A$$$. This divides the area into several smaller areas. The perimeter path of each new area consists of one aisle of $$$P_A$$$ and two newly created aisles. The area lies inside the triangle formed by these three aisles and it contains no other aisles. The scheme is applied repeatedly until no new area can be created.

The Manager wants to assess the profitability of his design. For this purpose, he defines a socalled highly-profitable configuration, which is a maximum-size set of machines with a property that each pair of machines in the set is directly connected by an aisle.

The manager is specifically interested in three profitability parameters. The first profitability parameter is the size of one highly-profitable configuration. The second profitability parameter is the number of highly-profitable configurations that exist in the design. The third profitability parameter is the number of slot machines which are part of at least one highly-profitable configuration.

You were recently promoted to captain and your troopers are now operating on a special mission in a storm. The battlefield is kind of extraordinary, it is located above the Arctic Circle on a huge ice floe. You coordinate the actions from the headquarters. There are many high-end computers keeping you up to date on the developments at the battlefield, which is modeled by the AI interface as a grid of unit squares. Each unit square is identified by its row and column index in the grid. Bigger rectangles, consisting of unit squares, are described by a pair of unit squares in the opposite corners of the rectangle. At the beginning, all squares are covered with ice.

There are two important types of information you receive from the computers:

0. Information about hits: Your enemy hit a rectangle described by unit squares $$$[x_1,y_1]$$$ and $$$[x_2,y_2]$$$. This rectangle is then immediately flooded by cold arctic water.

1. A query by your troopers: They ask whether it is possible to go from square $$$[x_1,y_1]$$$ to $$$[x_2,y_2]$$$ by a boat. The boat may be represented by a circle of radius $$$0.31416$$$. Note that the boat has to stay fully on water surface all the time and it is not allowed to leave the battlefield area.

Your troopers need your help! Can you guide them reliably?

After many long and exhausting years of war with dangerous overseas nations, our country has finally managed to get rid of almost all opposing forces and emerged victorious. Such a glorious victory shall be remembered and celebrated for many upcoming years. Because of that, our king has decided to declare the day of the victory as a public holiday, when a victory parade is to be held. During the parade, the king accompanied by his troops will start from his palace and subsequently visit all cities in the country.

The king and his suite will travel by a new type of environment-friendly electric helicopter that has a disadvantage of a relatively short operating radius. The king has requested you and your advisors to build helipads on some farm fields and in all the cities, so that each city is reachable from his palace by a sequence of short flights between helipads. However, building the helipads and the supporting infrastructure is costly. It is thus important to minimize the number of farm fields on which helipad construction takes place.

In addition, due to specific helicopter design, the king and his troops need to move in a special pattern which may influence the number and the locations of helipads.

You are given a rectangular grid map of the country, which consists of tiles of farm fields, tiles of cities and a tile of the king's palace. Also, you are given the movement pattern of the helicopter — it may move either as a Rook, a Queen, a Bishop, a Knight, or as a King in the game of chess (see images for movement illustration). Your task is to determine the minimal number of farm fields tiles and city tiles which have to host a helipad, in order to fulfill the conditions specified above. The tile with the king's palace already has a helipad and does not need a new one.

First, for those who have never seen a lamp before, let's say it is a cuboid (box with rectangular faces) made of glass and filled with gas. All sides of a lamp have integer lengths. Once upon a time, our lecturer was sentenced for destroying lamps on a street. He must have gone somewhat crazy, as he thought some of the lamps were screaming at him in high-pitched voices.

In his beautiful mind, he followed a weird pattern. He only recognized and destroyed those lamps which had no square face and whose volume did not exceed a fixed value. Later, during a session with his doctor Clarice, he said he was very scared of large objects and of objects with too regular shapes.

Your task is to count all possible shapes matching lecturer's conditions.

Agent Sue Thomas and her son are looking for trinities in a grid. The word trinity is a neologism referring to a particular triangular (as the morpheme "tri" suggests) shape composed of cells in the grid.

Each trinity is a result of taking a square-shaped area of the cells and removing all cells that lie either above or below one of the two diagonals of the area. The diagonal may be either the main diagonal (southeast-northwest direction) or the main antidiagonal (southwest-northeast direction). A valid trinity consists of at least three grid cells and all its cells contain the same character.

In the physics class, each student has to complete a semestral work. You chose to work on an experiment which examines the properties of rare unstable particles, known as red balls. A red ball particle travels in a straight line until it is reflected by a mirror.

Our experiment can be thought of as several one-reflection mirrors, which can be treated as (potentially infinite) lines in a 2D plane. A one-reflection mirror is made in a way that it reflects a particle only for the first time the mirror is hit by the particle, no matter which direction the particle is coming from. After that, the mirror will always let that particle pass through.

When reflecting, the angle of the line along which the particle approaches the point of reflection is equal to the angle of the line along which the particle leaves the point of reflection, when both angles are measured to the normal of the reflecting mirror. The particle disintegrates if it gets to a crossing point of two mirrors which would otherwise reflect it.

The mirror layout was created by your teacher and your task is to find directions which cause a particle launched from its given starting location to reach a specified final location.

Two players, Petyr and Varys, play a game in which players remove stones from $$$N$$$ piles in alternating turns. Petyr, in his turn, can remove at most $$$A$$$ stones from any pile. Varys, in his turn, can remove at most $$$B$$$ stones from any pile. Each player has to remove at least one stone in his turn. The player who removes the last stone wins.

The game has already started and it is Petyr's turn now. Your task is to determine whether he can win the game if both he and Varys play the game in the best possible way.