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Pick up sticks Description: Pick up sticks is a fascinating game. A collection of coloured sticks are dumped in a tangled heap on the table. Players take turns trying to pick up a single stick at a time without moving any of the other sticks. It is very difficult to pick up a stick if there is another ...
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Picking Up Steam Description: Meteorologist Wendy Wynne Blose and geologist Maddy Morfik are in the Porous Mountains of Central Asia, studying mild eruptions of superheated smoke and steam from below ground. Unlike more dramatic eruptions such as the one that blew the top off Mount St. Helens in $1980$...
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Picking Up the Dice Description: Two players are playing a game with a set of $K$ six-sided dice. One player calls out a number in the range $K\ldots 6K$ and the other tries to roll that number. After the first roll, the player is allowed to pick up any number ($0\ldots K$) of dice and re-roll them. G...
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Pickle Clicker Description: Approximately six months ago, the critically acclaimed game Pickle Clicker was released. After spending countless hours on the game, Rasmus has decided to take things to the next level: he plans to speedrun the game. The goal of Pickle Clicker is to collect as much of the c...
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Picture Day Description: The school for applied astrology has decided to take pictures of all their students. With moon phases being what they are, combined with Saturn’s position relative to Venus, they are forced to focus on cost. In order to minimise cost, they have asked three photographers for th...
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Pie Description: My friends are very annoying and if one of them gets a bigger piece than the others, they start complaining. Therefore all of them should get equally sized (but not necessarily equally shaped) pieces, even if this leads to some pie getting spoiled (which is better than spoiling the par...
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Piece It Together Description: Tom has developed a special kind of puzzle: it involves a whole bunch of identical puzzle pieces. The pieces have the shape of three adjoint squares in an L-shape. The corner square is black, the two adjacent squares are white. The puzzler is given a pattern of black and whi...
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Piece of Cake Description: Alice received a cake for her birthday! Her cake can be described by a convex polygon with $n$ vertices. No three vertices are collinear. Alice will now choose exactly $k$ random vertices ($k{\ge }3$) from her cake and cut a piece, the shape of which is the convex polygon define...
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Piece of Cake! Description: It is Greg’s birthday! To celebrate, his friend Sam invites Greg and two other friends for a small party. Of course, every birthday party must have cake. Sam ordered a square cake. She makes a single horizontal cut and a single vertical cut. In her excitement to eat cake, Sam ...
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Pieces of Parentheses Description: You are teaching a class in programming, and you want to cover balanced parentheses. You’ve got a great visual aid, a sign with a very long, balanced string of parentheses. But, alas, somehow, your visual aid has been broken into pieces, and some pieces may be missing...
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Pig Latin Description: Ancay youyay eakspay igpay atinlay? (Can you speak pig latin?) If you can’t, here are the rules: * If a word begins with a consonant, take all of the letters before the first vowel and move them to the end of the word, then add ay to the end of the word. Examples: pi...
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Pigstep Description: Steve and Alex have a challenge for you. They are each thinking of an integer between $1$ and $500$, and your task is to guess their numbers. If you can guess both numbers in $10$ guesses, they’ll give you a rare prize: a Pigstep music disc. Are you up to the challenge? To make it...
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Pillow Stacking Description: Eleanor has recently lost her favorite pillow. Being very particular about her pillows, she categorizes pillows by their softness. The pillow she lost had softness $C$, and she will be unable to sleep without this exact softness. While she has access to some other types of ...
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Pinball Description: Maria is quite addicted to pinball. She can shoot the ball to any position at the top of the board, but she cannot predict where the ball will end when it falls down, because it hits many bumpers on its way down. She decided to model the pinball table as line segments and assume t...
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Ping Pong Tournament Description: Rulls is a true enthusiast of PO (the Pingis Olympiad). He loves following the matches and cheering on his favorite players as they compete. PO is a table tennis tournament with an exciting format. The tournament begins with $N$ players, where each player has a starti...
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Pinned Files Description: You recently discovered a new feature in Visual Studio — pinned files! You have no idea what pinned files are good for — you could hear a pin drop when you asked your friends about it — but that doesn’t stop you from pinning and unpinning files all day and night. Visual Studi...
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Pinni Frændi Description: Hannes was ordering a pizza, but had forgotten his PIN, as per usual. Luckily he has it saved in the contact Pinni Frændi on his phone. While he was still looking at that contact he got the confirmation number from Panucci’s Pizza. Surprisingly enough his PIN number and the co...
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Pipe Rotation Description: The four ninja turtles: Leonardo, Donatello, Michelangelo, and Raphael are seeking a new home in Manhattan, New York City. The turtles don’t like sudden dead-ends in their home. Fortunately, the government recently installed a new sewage system where pipes can be rotated! The...
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Pipe Stream Description: In the possible event that somebody somewhere decides they want some Flubber, they would like to know how quickly it will flow through the pipes. Measuring its rate of flow is your job. You have access to one of the pipes connected to the network. The pipe is $l$ meters lo...
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Pipes Description: Kažimír got an important job at the plumbing department. He needs to connect certain pairs of locations by pipes. However, as the department’s budget is very low, they have only two types of pipe parts: straight and “elbow” (L) shaped ones, all of the same size. Moreover, each pipe p...
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Pipes Description: After writing a solver for the “moveable maze” game last week, you have grown tired of it. After all, you already know the optimal solution. To entertain yourself, you find another puzzle game called “Pipes”, and play that for a while. On one puzzle, you have not been able to find a ...
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Pipes Description: The construction of office buildings has become a very standardized task. Pre-fabricated modules are combined according to the customer’s needs, shipped from a faraway factory, and assembled on the construction site. However, there are still some tasks that require careful planning, ...
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Pirate Chest Description: Pirate Dick finally had enough of fighting, marauding, theft, and making life miserable for many on the open seas. So he decided to retire, and he found the perfect island to spend the rest of his days on, provided he does not run out of money. He has plenty of gold coins now,...
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Pitch Performance Description: After a recent disaster at the Easter party karaoke, you are working on improving your singing. To gauge how well you are doing, you would like to measure how much the pitch and timing of your singing differs from the target melody you were trying to perform. We model th...
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Pivot Description: An $O(n)$ Partition algorithm partitions an array A around a pivot element (pivot is a member of A) into three parts: a left sub-array that contains elements that are $\leq $ pivot, the pivot itself, and a right sub-array that contains elements that are $>$ pivot. A Partition al...
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Pixelated Description: An LCD panel is composed of a grid of pixels, spaced $1$ alu (“arbitrary length unit”) apart both horizontally and vertically. Wires run along each row and each column, intersecting at the pixels. Wires are numbered beginning with $1$ and proceeding up to a panel-dependent ma...
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Pizza Crust Description: George has bought a pizza. George loves cheese. George thinks the pizza does not have enough cheese. George gets angry. George’s pizza is round, and has a radius of $R$ cm. The outermost $C$ cm is crust, and does not have cheese. What percent of George’s pizza has cheese? ## Inpu...
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Pizza Cutting Description: Pizzas are usually cooked as a circle, and are often cut “triangularly” along lines moving out from the center, as shown on the left in Figure 1 below. However, some heretical people prefer that the pizza be cut into rectangular pieces, as shown on the right in the figure: A...
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Pizza Delivery Description: Your Irish pizza and kebab restaurant is doing very well. Not only is the restaurant full almost every night, but there are also an ever increasing number of deliveries to be made, all over town. To meet this demand, you realize that it will be necessary to separate the deli...
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Pizza Hawaii Description: You are travelling in a foreign country. Although you are also open to eat some regional food, you just cannot resist after you have found an Italian restaurant which offers pizza. Unfortunately the menu is written in the foreign language, so the list of ingredients of the piz...
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Pizza Party! Description: You are co-organizing a computer science conference, and you are in charge of a pizza party for the conference guests. Each guest holds preferences over combinations of toppings, and guests are seated in groups by table in the conference center ballroom. One pizza is served to...
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Pizza Problems Description: Me and my friends are ordering a big pizza to share. As you can imagine this is quite complicated, since everyone has different wishes about what should be on the pizza. For instance Gunnar wants bananas on the pizza, Emma doesn’t want bananas but wants olives, Marc wants th...
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Pizza slicing Description: Tonight you will be at one of the regular GAUSS sky observation nights, and as always there’s free pizza. The $N$ participants have ordered a single giant $A\times B$ rectangular pizza that you will have to share between you. To break the monotony of looking at differently si...
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Pizzastrengur Description: Today is game day at Tommi’s pizzas so Georg and his friends are going to order some pizzas. The game starts with the employees at Tommi’s pizzas choosing a secret passcode, consisting only of the letters {“P”, “I”, “Z”, “A”}, and telling the customer the length of th...
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Pizzubestun Description: Nonni has asked his guests what kind of pizza they would like, so he knows how many pizzas need to be ordered. Nonni wishes to pay as little as possible for the pizzas and asks you for help. Given the pizzas he wishes to order and the price of each pizza you need to figure out whi...
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Plan Your Guesses Description: After the Deer God heals Ashitaka in the forest, Ashitaka gives San a challenge to distract himself. He tells San that his favorite number is between $1$ and $n$, and wants her to guess it. Anytime San guesses a number, she will be told that the guess is correct, or that it ...
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Plane Ticket Pricing Description: Plane ticket prices fluctuate wildly from one week to the next, and their unpredictability is a major source of frustration for travellers. Some travellers regret buying tickets too early when the prices drop right after they purchase the tickets, and some travellers r...
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Planes, Trains, but not Automobiles Description: Being a traveling salesman is tough work. Per is one such salesman and would like to find an efficient way to visit all the cities in a foreign country exactly once. Per defines efficiency in a peculiar way as Per hates flying on planes. Even worse, he abso...
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Planet Destruction Description: Darth Vader is back to his favourite hobby: destroying planets (or their population to be precise). He just discovered that the rebel leadership has gathered at a planet named Watooine, so he must act quickly to eliminate the threat. Unfortunately, the Empire is in a pre...
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Planet Hopping Description: As part of the Constellation Commerce and Planetary Connections company (CCPC), you are tasked with mining resources from planets in the Milky Way. As a part of this task, you are given a spaceship that can travel to different planets in the Milky Way, as well as a device th...
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Planetaris Description: Oh, no. Atli has just painted himself into a corner. He has been playing the video game Planetaris and decided to start a war against his friend Finni. The downside is Finni had a considerably larger spaceship fleet than Atli expected. Atli now needs your help to turn the tide, ...
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Planetary Grid Description: Yraglac wants to get around within his gated community on Mars, Cossin. The Cossin community grounds are precisely – but strangely – shaped. Yraglac has spent many hours studying the grounds and has figured out the logic behind the landscaping. If you put the landscaper’s h...
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Planete Description: The European Space Agency has $N$ telescopes observing distant planets. They have classified $M$ different types of events. They noticed that every day on each planet there is exactly one type of event. They measure events in days because they discovered each event lasts at least one ...
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Planets Description: The spaceport likes collecting data on its visitors. We know that there are $p$ nearby planets that generate all traffic in and out of the spaceport. Each planet contains $n$ species and every ship contains $m$ passengers of a particular species. No two planets contain the same...
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Planina Description: Mirko and Slavko are filming a movie adaptation of the popular SF novel “Chicks in space 13”. The script requires them to present a lot of different worlds so they decided to film the entire movie in front of a green screen and add CGI backgrounds later. Mirko heard that the best w...
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Plankton Food Description: Only few ZOOs can afford to cultivate all types of plankton food they need to feed to various sea vertebrates and invertebrates which live in their voluminous aquariums. Some ZOOs might be from time to time in a short supply of a particular kind of plankton food which is mome...
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Planting Trees Description: Farmer Jon has recently bought $n$ tree seedlings that he wants to plant in his yard. It takes $1$ day for Jon to plant a seedling1, and for each tree Jon knows exactly in how many days after planting it grows to full maturity. Jon would also like to throw a party for his fa...
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Plants vs Bad Guys Description: Oh no! There are a bunch of bad guys creeping around on Mikael’s lawn, and they all want to attack Mikael! Luckily, Mikael has planted several peashooters on his lawn that can protect him from the bad guys. Mikael’s lawn consists of $N$ rows, where Mikael has planted $R...
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Platform Placing Description: The city of Atlantis is making an above-water section. They are doing so by building floating platforms that are anchored at their centers to foundation points that lie in a straight line at the bottom of the ocean. Each platform is a fixed-width rectangle aligned to the f...
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Platforme Description: A level is being designed for a new platform game. The locations of the platforms have been chosen. Contrary to popular opinion, platforms can’t float in the air, but need pillars for support. More precisely, each of the two ends of the platform needs to be supported by a pillar ...
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Playfair Cipher Description: The Playfair cipher is a manual symmetric encryption technique and was the first digraph substitution cipher. The scheme was invented in 1854 by Charles Wheatstone, but bears the name of Lord Playfair who promoted the use of the cipher. The Playfair cipher uses a $5$ by $5...
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Playground Description: George has $K \le 20$ steel wires shaped in the form of half-circles, with radii $a_{1}, a_{2}, ..., a_{K}$. They can be soldered (connected) at the ends, in any angle. Is it possible for George to make a closed shape out of these wires? He does not have to use all the wires...
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Playing the Slots Description: The small nation of Erratica prides itself on defying conventions established by the more “boring” countries around the world. One of their more obvious distinctions lies in the design of their coinage. Believing that a person should be easily able to identify the value a...
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Playing with Geometry Description: Dynamic geometry software can help students understand transformation geometry as it allows to visualize the effect of transformations on a shape. Alice is learning the elementary transformations – slides, flips and turns – or, more formally, translations, reflections...
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Playing with Numbers Description: You have a list of $N$ numbers, each of the form $2^ a3^ b$ for some non-negative integers $a$ and $b$. You want to perform $N-1$ operations on these numbers. Each operation acts on two numbers $X$ and $Y$ of your choice from the list, replacing them with a new number ...
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Playlist Description: In order to give the show a sense of continuity, you have decided that consecutive songs you play should have some detail in common. For instance, if you start by playing Björk’s song “Pagan Poetry”, you could continue with next playing matmos’ track “Enigma Machine for Alan Turin...
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Please, Go First Description: You are currently on a skiing trip with a group of friends. In general, it is going well: you enjoy the skiing during the day and, of course, the après-skiing during the night. However, there is one nuisance: the skiing lift. As always, it is too small, and can only serve ...
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Plotting Polynomials Description: Graphical calculators have become popular among high school students. They allow functions to be plotted on screen with minimal efforts by the students. These calculators generally do not possess very fast processors. In this problem, you are asked to implement a metho...
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Plus Minus Description: Matthew the physicist studies the quantum electro-dynamics of a silicon-based rectangular microchip. The microchip consists of a very large $N \times M$ grid of electrons. Each electron has either positive (up) or negative (down) spin, denoted by $+$ and $-$ respectively. Matth...
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Podnizovi Description: You are given an array of integers of length $N$. Let $s_1, s_2, \ldots , s_ q$ be the lexicographically sorted array of all its non-empty subsequences. A subsequence of the array is an array obtained by removing zero or more elements from the initial array. Notice that some subs...
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Poetry Tower Description: Alice and Bob are young, aspiring poets. Alice likes variation in poems, so she only gets impressed by a poems that do not contain repeated words (and of course, the words used have to be real words). Bob has found $N$ six-sided blocks where each side has a letter or a wildca...
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Point in Polygon Description: ## Input Input contains several test cases. Each test case begins with an integer $n$ ($3 \leq n \leq 1000$). Then follow the $n$ vertices of a simple polygon, one per line, each of the form $x\ y$. The points may be given in either clockwise or counterclockwise o...
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Pokeball Fever Description: Pokemon Go has become a recent trend. Zapray becomes immediately addicted to the game after it comes out. Zapray’s Pokemon journey begins with a bag of $100$ Pokeballs. Each time he encounters a Pokemon, he keeps throwing Pokeballs to the Pokemon until he either catches the ...
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Pokechat Description: The year is 2011, and you and your friend each got a copy of the newly released Pokemon Diamond and Pearl games. You enjoy playing the game together, but you can’t always be together and talk while you’re playing because your mom makes you come home after dark. This would be fine,...
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Pokegene Description: Professor Oak, the friendly neighbourhood Pokémon Professor, has been getting a lot of questions about Pokémon ancestors lately. One of the things a lot of Pokémon trainers are curious about is finding common ancestors of Pokémon in their Pokédex. In particular, each Pokémon Trai...
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Pokemon Go Go Description: The program assumes that the user is in a city where travel is restricted to moving only in the north–south and east–west directions. The program also assumes that all poké stops are on the intersection of two roads. For example, consider a case where the application finds f...
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Poker Hand Description: You are given a five-card hand drawn from a standard $52$-card deck. The strength of your hand is the maximum value $k$ such that there are $k$ cards in your hand that have the same rank. Compute the strength of your hand. ## Input The input will consist of a single line, with fi...
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Poklon Description: Mirko got a set of intervals for his birthday. There are many games he can play with them. In one of them, Mirko must find the longest sequence of distinct intervals such that each interval in the sequence is in the set and that each interval contains the one that follows in the seq...
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Pokémon Ice Maze Description: You are hired as a level designer for the next Pokémon series, with games called Ice and Fire. For the first of these two games, players have to get through a maze in an icy cave. The cave is represented as a grid, with each square of the grid being either ice, gravel or a...
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Pole Colouring Description: All the street lights of lineland should be coloured red, blue, green, yellow, black or white. Red and blue paint comes in packages that suffice for two poles, and there’s some left over blue paint in storage. Thus an even number of poles must be red and an odd number must b...
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Police Description: Librarian Jurica has $N$ shelves in his library, and each shelf can contain $M$ books. Jurica is a good librarian so he decided to make an inventory in the library and, if it’s necessary, return the books that aren’t in their place to their right place. He moves the books in the...
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Polish Notation Description: One way of representing mathematical expressions is with Polish (or prefix) notation, initially developed by Jan Łukasiewicz (1878–1956) for use in logic. In this notation, the operator appears before its operands. For example, rather than writing using so-called infix not...
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Political Development Description: A certain political party with $N$ members wants to develop some brand new politics. In order to do so, the party plans to make a committee for new political development. Clearly, the best politics is developed when all committee members disagree with each other, and ...
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Pollution Solution Description: As an employee of Aqueous Contaminate Management, you must monitor the pollution that gets dumped (sometimes accidentally, sometimes purposefully) into rivers, lakes and oceans. One of your jobs is to measure the impact of the pollution on various ecosystems in the water...
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Polly Gone Description: Polly, the programming parrot, has escaped. As Johan is in desperate need for some programming skill, he wants to find her. The only clue that Johan has about the escape route is that Polly flew into his kitchen. Johan has $N$ boxes on his kitchen table, and each box has a cert...
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Polygon Area Description: Compute the area of a simple polygon, and the direction (clockwise or counterclockwise) in which its vertices are given. ## Input Input contains up to $25$ test cases. Each test case begins with an integer $n$ ($3 \leq n \leq 1\, 000$). Then follow the $n$ vertices o...
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Polygon Game Description: Jack and Jill are playing a game they call Polygon Game. In this game they start with a convex polygon and then take turns dividing the polygon into smaller parts. They divide the polygon by drawing a straight line from one point on the border of the original polygon to anothe...
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Polygonal Puzzle Description: During last year’s ACM ICPC World Finals in Marrakesh, one of the judges bought a pretty wooden puzzle depicting a camel and palm trees (see Figure 1). Unlike traditional jigsaw puzzles, which are usually created by cutting up an existing rectangular picture, all the piece...
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Polygraph Description: On the distant isle of Googlia, there are two cities, Truthtown and Liarville. People from Truthtown always tell the truth and people from Liarville always lie. While exploring Googlia, you have run across a group of N inhabitants, and you want to figure out which city each one c...
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Polyline Simplification Description: Mapping applications often represent the boundaries of countries, cities, etc. as polylines, which are connected sequences of line segments. Since fine details have to be shown when the user zooms into the map, these polylines often contain a very large number of se...
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Polynomial Multiplication 1 Description: ## Input Standard input begins with an integer $T \le 100$, the number of test cases. Each test case consists of two polynomials. A polynomial is given by an integer $1 \le n \le 255$ indicating the degree of the polynomial, followed by a sequence of integers ...
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Polynomial Multiplication 2 Description: ## Input Standard input begins with an integer $T = 1$, the number of test cases (this is not a typo, there is always exactly one test case). Each test case consists of two polynomials. A polynomial is given by an integer $1 \le n \le 131\, 071$ indicating the...
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Polyomino Powers Description: ## Input One line with two positive integers $h, w \leq 10$. Next follows an $h \times w$ matrix of characters ‘.’ or ‘X’, the ‘X’s describing a polyomino and ‘.’ space. ## Output A $k$-power with $2 \leq k \leq 5$ copies of a smaller polyomino: Output a $h\time...
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Pong Tournament Description: The Swedes wanted to arrange a ping pong tournament among the $n$ high school students in the nation. Since Sweden is the fairest country on earth, they wanted to ensure that the ranking of all the players is perfectly fair. Therfore, they decided that every high school...
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Pony-less Express Description: Howdy, pardner! You’ve been put in charge of mail delivery in these here parts. Your headquarters is in Capital City, and Miss Penelope has asked us to tell all of the local farmsteads about her fashionable doin’s. The roads hereabouts were built so that each farmstead co...
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Pop! Description: A large number of helium balloons have been set up as part of a carnival game. Players shoot a pellet gun into the balloons and earn a point for every balloon they pop. Each balloon is spherical and is tied to a string at the bottom. Each string is taped to a table top at a particular...
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Popcorn Description: The culmination of the Galactic Popcorn Festival G-Pop is undoubtedly the competitive eating event. First, the participants are separated into four groups of equal size. Then each participants competes against every other member of the same group; the goal is to eat half a bag of ...
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Poplava Description: Mirko dreamt of a histogram last night that consists of $N$ columns. Each column is one meter wide and the heights of the columns in meters are $h_1, h_2, \ldots , h_ N$. The capacity of a histogram is the maximal amount of water that a histogram can hold so that the configura...
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Popločavanje Description: Mirko’s ASCII street is made of $N$ lowercase letters of the English alphabet. The city government occasionally replaces the tiles in the street. However, the letter tiles are in high demand, so the government has only $M$ different tile patterns available. The $i$th tile pat...
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Popular Vote Description: ## Input The first line of input contains a single positive integer $T \leq 500$ indicating the number of test cases. The first line of each test case also contains a single positive integer $n$ indicating the number of candidates in the election. This is followed by $n$ line...
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Popularity Contest Description: It’s not about what you know, but who you know. A famous saying – but is it true? You start looking around yourself, pondering what lies behind the success of some of your friends. Could it be that life is just one big popularity contest? Are all the hours you spend lear...
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Pork Barrel Description: Winning the election was simpler than you expected: it was enough to promise to finally build a good quality, country-wide road infrastructure, of course without crippling the budget…Your happiness did not last long, however: it seems, that the citizens have found a way to actu...
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Porridge Boxes Description: Julia loves porridge. So much in fact, that if she had the possibility she would eat it all day, any day. However, due to a busy schedule, she does not have time to make porridge every day. But on days when she does have time to make porridge, she can make a big batch in her...
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Portal Description: You think it would be funny to prank your best friend by placing them on cell $(0, 0)$ of an infinite grid of coloured cells. The friend then moves around the grid indefinitely, one step at a time, always moving to one of the four adjacent cells. $N$ of the cells on the grid co...
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Positive Divisors Description: Given a positive integer $N$, compute all of its positive divisors. ## Input The first and only line contains the integer $N$ ($1 \le N \le 10^{15}$). ## Output Output all the divisors of $N$, in ascending order, one per line. **Sample Input 1** **Sample Output 1** ```...
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Poslozi Description: “Arrange” is a planetary popular Flash game. In “Arrange” the player is given a permutation of numbers $1$ to $N$ and a list of allowed swaps. He then has to perform a sequence of swaps that transforms the initial permutation back to the ordered sequence $1,2,3,4,5, \ldots , N$. In or...
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Postal Delivery Description: The postal service is interested in cutting costs as an alternative to raising the postage rates. One way to do this is by minimizing the distance traveled when delivering mail from the post office to all the required locations and returning to the post office. It may be th...
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Posterize Description: Pixels in a digital picture can be represented with three integers in the range $0$ to $255$ that indicate the intensity of the red, green, and blue colors. To compress an image or to create an artistic effect, many photo-editing tools include a “posterize” operation which wo...
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Pot Description: The teacher has sent an e-mail to her students with the following task: “Write a program that will determine and output the value of $X$ if given the statement: and it holds that $\mathit{number}_1$, $\mathit{number}_2$ to $\mathit{number}_ N$ are integers, and $\mathit{pow}_1$, $...
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Potion Commotion Description: You are incredibly close to becoming the Pokemon champion, but one final obstacle stands in your way: the Elite Four. Throughout your journey, you raised your starter Pokemon to a high enough level, and out of a show of loyalty to your favorite Pokemon, you vowed to never ...
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