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Report Card Description: After coming to college from high school, you discovered something absolutely tragic. The way that GPA is calculated is completely different than what you are used to. Apparently, pluses and minuses are treated very differently than they were before, according to the following ...
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Rerouting Rapids Description: The famous East Manitoba river that flows through Edmonton consists of $N$ rapids. The river is a popular rafting destination, with many visitors coming to the river each summer. When a visitor enters a rapid $i$, the current will take them to some unique other rapid $...
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Research Productivity Index Description: Angela’s research group has a unique way of evaluating the success of paper submissions. They use the research productivity index, defined as $a^{a/s}$, where $s$ is the total number of papers submitted, and $a$ is the number of papers that are accepted by the c...
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Reservoir Description: A big reservoir was built on Red River using a dam. Assume that the reservoir is a rectangular box with unit length width. The reservoir consists of many tanks. An example a cross section of an empty reservoir along its length and height dimensions is shown in the picture below: ...
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Reset Description: Li and Xiao are involved in a supernatural event in which a crisis will happen at a fixed time in the future. They must complete a number of tasks to prevent the crisis from happening. Once they start a task, they must finish it before switching to another task. Tasks can be complete...
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Reseto Description: The sieve of Eratosthenes is a famous algorithm to find all prime numbers up to $N$. The algorithm is: * Write down all integers between 2 and $N$, inclusive. * Find the smallest number not already crossed out and call it $P$; $P$ is prime. * Cross out $P$ and all...
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Restaurant Bribes Description: Several participants in programming competitions end up founding startup companies. Two of them created Eeet, a social restaurant rating system. Each member of Eeet can rate a restaurant with an integer score from $0$ to $10$ or not give a score at all. The score that a p...
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Restaurant Opening Description: It is said that the three most important factors for determining whether or not a business will be successful are location, location, and location. The Incredible Cooks Preparing Cuisine are opening a new restaurant in the International City Promoting Cooking, and they h...
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Restaurant Orders Description: A friend of yours who is working as a waiter has a problem. A group of xkcd-fans have started to come to the restaurant and order food as in the comic strip below. Each order takes him a lot of time to figure out, but maybe you can help him. ## Task You are to write a progr...
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Restaurant Ratings Description: The restaurants in each city are ranked as follows. First, sum up the ratings given by all the critics for a restaurant. A restaurant with a higher total sum is always better than one with a lower total sum. For restaurants with the same total sum, we rank them based on ...
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Restoran Description: In Croatia there are $N$ cities connected by $E$ two-way roads. Two large food chains have recently reached an agreement on market sharing. In the middle of each road, exactly one chain will be given rights to build a restaurant. To ensure the market is shared fairly, each city must ...
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Restroom Monitor Description: Irma P. Freely (yes, we’ve hit a new low) is in charge of the bank of restrooms at the Rest Pit truck stop. Every so often a tour bus stops by and a load of passengers gets off to use the restroom. Irma has a set of $n$ single-stall restrooms she can allocate to people. Ev...
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Retribution! Description: The coaches in a certain regional are fed up with the judges. During the last contest over $90$% of the teams failed to solve a single problem—in fact, even half the judges found the problems too hard to solve. So the coaches have decided to tar and feather the judges. They kn...
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Return of the Jedi Description: Luke Skywalker races through the forest on a speeder bike, trying to outrun a patrol of Imperial scouts on Endor. A small moon near a new Death Star, Endor is covered by dense foliage and a thick forest of ancient towering trees. The speeder bike, which Luke stole from a...
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Reverse Rot Description: A very simplistic scheme, which was used at one time to encode information, is to rotate the characters within an alphabet and rewrite them. ROT13 is the variant in which the characters A-Z are rotated 13 places, and it was a commonly used insecure scheme that attempted to “hid...
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Reversed Binary Numbers Description: Yi has moved to Sweden and now goes to school here. The first years of schooling she got in China, and the curricula do not match completely in the two countries. Yi likes mathematics, but now... The teacher explains the algorithm for subtraction on the board, and Y...
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Reversibly Cyclic Strings Description: A string $t$ is a Cyclic Substring of a string $s$ if there is some rotation of $s$ such that $t$ is a substring of that rotation of $s$. For example, if $s$ is fatcat, then atc and atf are both Cyclic Substrings of $s$. However, act is not a Cyclic ...
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Reversing Roads Description: You work for the city of One-Direction-Ville. The city mandates that every road in its limits be one direction only. You are evaluating proposals for a new subdivision and its road network. One problem you’ve observed in some early proposals is that it is impossible to get ...
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Revision Control Description: Emil is the treasurer of a nonprofit organization, and he has received $N$ requests to record a specific receipt. However, the issue is that some receipts may have been submitted multiple times but by different individuals. Your mission is to help Emil figure out what he ...
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Rečenice Description: Once upon a time in a land far far away, inhabited only by math students, Iva and Vedran were discussing self-explanatory sentences. A part of these sentences is exactly one number and it is equal to the total number of letters in the sentence. Some examples are: “This sentence ha...
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Rhinoceros Beetle Description: ## Rules of Texas Hold’em Poker Five cards (the community cards) are shared by all the players and each player holds 2 other cards that may be combined with the community cards, meaning that each player has 7 cards available, out of which 5 cards are chosen to make the b...
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Rhombinoes Description: In the game of Rhombinoes, you have a board made up entirely of equilateral trianges (see the image), some of which are “live” and some are “dead”. Your goal is to place down as many rhombinoes (“rhombus”-shaped pieces) as possible on the board. Each rhombino should exactly cove...
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Rhyme Power Description: Rhymes are complicated. The extent to which one word rhymes with another depends on how similar they sound; but if they are too similar then they aren’t considered a rhyme at all. Karim has a huge list of $N$ words and wants to determine the maximally rhyming pair of words....
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Rhyming Slang Description: Rhyming slang involves replacing a common word with a phrase of two or three words, the last of which rhymes with the original word. For example, * replacing the word “stairs” with the rhyming phrase “apples and pears”, * or replacing “rotten” with the phrase “bales of ...
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Rhythm Flow Description: You are designing a scoring algorithm for the new hit rhythm game Rhythm Flow where players must press a button in time to the music. During a round of Rhythm Flow, there are points in time when a visual indicator flashes on the screen. Players are expected to press the button ...
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Ribbon Road Description: Christie is an ant that likes to crawl on a closed ribbon (i.e., it forms a polygon with start and end points attached). Her owner, Cindy would place a sugar cube on the ribbon for her to enjoy. Cindy always puts the sugar cube on the exterior side of the ribbon, so Christie wa...
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Rice judge Description: Sir Dorian loves to eat rice. He always eats lunch at the rice palace, where every day there are $N$ portions of rice. He has eaten rice for solng that he can determine the amount of rice grains in a portion, in addition to the size of the rice grains in nanometers simply by loo...
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Ricochet Robots Description: A team of up-to four robots is going to deliver parts in a factory floor. The floor is organized as a rectangular grid where each robot ocupies a single square cell. Each robot is represented by an integer from 1 to 4 and can move in the four orthogonal directions (left, ri...
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Riddle of the Sphinx Description: In this problem, you meet a different sphinx who gives you a somewhat reversed riddle: “How many legs do an axex, a basilisk, and a centaur have?” While you recognize these as creatures from Egyptian and Greek mythology, you have no clue how many legs each has (except ...
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Ride-Hailing Description: Jamal owns a new ride-hailing business. His business operates in the following way: At least $12$ hours before a trip is desired, a customer orders a trip from a starting location to an ending location to take place at a specified starting time. Although Jamal has considered r...
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Right-of-Way Description: A startup company tries to develop software for self-driving cars. One of the surprisingly difficult problems turns out to be right-of-way at an intersection. The local traffic regulations prescribe that your vehicle must yield the right-of-way if one of the following two cond...
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Riječi Description: One day, little Mirko came across a funny looking machine! It consisted of a very very large screen and a single button. When he found the machine, the screen displayed only the letter A. After he pressed the button, the letter changed to B. The next few times he pressed the button,...
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Rimski Description: Using roman numerals the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written as ‘I’, ‘II’, ‘III’, ‘IV’, ‘V’, ‘VI’, ‘VII’, ‘VIII’, ‘IX’. The numbers $10, 20, 30, 40, 50, 60, 70, 80, 90$ are written as ‘X’, ‘XX’, ‘XXX’, ‘XL’, ‘L’, ‘LX’, ‘LXX’, ‘LXXX’, ‘XC’. Any number smaller than $10...
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Rings Description: Little John is in big trouble. Playing with his different-sized (and colored!) rings and glue seemed such a good idea... However, the rings now lay on the floor, glued together with something that will definitely not come off with water. Surprisingly enough, it seems like no rings ar...
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Rings Description: Dee Siduous is a botanist who specializes in trees. A lot of her research has to do with the formation of tree rings, and what they say about the growing conditions over the tree’s lifetime. She has a certain theory and wants to run some simulations to see if it holds up to the evide...
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Ringworld Description: The world is actually neither a disc or a sphere. It is a ring! There are $m$ cities there, conveniently called $0,1,2,\ldots ,m-1$, and arranged on the ring in the natural order: first $0$, then $1$, then $2$, ..., then $m-1$, and then again $0$ (as the world is a ring, remember...
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Rinse Description: Granny can upend the cream can, thus pouring its contents on the ground. But due to surface tension and the shape of the can a small amount of the content remains, even after the can is upended. Granny has a barrel full of rain water that she can use to rinse the can several times. E...
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Rise and Fall Description: A number is said to Rise and Fall if the decimal representation can be broken up into two parts (possibly empty) where the first part has digits in nondecreasing order and the second part has digits in nonincreasing order. Note that a proper integer may not have any leadi...
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Risk Description: Risk is a board game played on a world map. This world is divided into regions by borders. Each region is controlled by a player (either you or one of your opponents). Any region that you control contains a positive number of your armies. In each turn, you are allowed to move your armies...
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Risky Lottery Description: Prof. Peter decided to surprise his class by organizing a new lottery with a very peculiar system. He starts by announcing a small positive number $M$. Afterwards, each student is going to secretly write an integer from the set $\{ 1, ..., M\} $ on a slip of paper that i...
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Road Times Description: Ubol Narongdid is the founder of a brash new startup company called Special D-Liver-E. She wants to corner the market on overnight deliveries of organs between hospitals in the Phuket area. For scheduling purposes it is important to have accurate estimates for the times to perfo...
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Road To Savings Description: Pat Wholes is in charge of road maintenance in Capitol City, and boy do those roads need maintenance. The road conditions are so poor that accidents have become almost a daily event, and the surviving public is in an uproar. Now while Pat would love to pave every road in th...
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Road Trip Description: Tony is going on a road trip. Because gas is so expensive, Tony wants to carefully choose the gas stations he refuels at to save as much money as possible. For simplicity, assume Tony is driving on a straight road from left to right. There are $n$ gas stations along the road, where ...
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Road Trip! Description: If you’ve done the practice exams, you probably remember Alice, the MPCS student who uses the power of algorithms to plan her mountain hikes. As it turns out, Alice also loves going on road trips! The way that Alice comes up with her road trip itinerary is governed by three simp...
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Road Work Description: Per is repairing roads. The job is concentrated on roads with one lane in each direction. Thus, when Per closes down the lane in one direction, all traffic has to go through the other lane. This is done by allowing only one direction of travel at any time. Per is often assigned t...
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Roaming Romans Description: Over time, the actual distance referred to as a “mile” has changed. The modern English mile is $5\, 280$ (modern) feet. The Roman mile is believed to have been about $4\, 854$ (modern) feet. Therefore a distance of $x$ English miles would correspond to $1\, 000 \cdot \fr...
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Robbers are Often Robbed Description: Mario finds himself in a wealthy neighborhood. There are $N$ homes on a street each with $S_ i$ stars where $i \in [1, N]$. He is given $K$ burlap sacks with an infinite capacity to hold stolen stars. He can take as many or as few stars as he’d like from ea...
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Robert Hood Description: Robert Hood, a less famous sibling of the Robin Hood, is fed up. Despite him being a young, talented archer he never seems to reach quite the same level as his legendary brother, and so he decided to come up with rules for a new archery contest, in which he will stand a better ...
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Robot Delivery Description: What a busy day! Tyko has been diligently preparing problems for an upcoming programming contest all day. He has been so busy that he forgot to eat lunch and now he is quite hungry! "What if I made some sort of robot-based food delivery system?" he thinks to himself. Of cour...
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Robot Protection Description: You are developing a new robotic guard that can wander around and look for suspicious activity in your home. The robot is small and not smart enough to maneuver by itself. Instead, you set up fixed-location beacons in the region you want it to protect. The robot is program...
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Robot Turtles Description: Robot Turtles is one of Theta’s favorite games. In this game, kindergarteners learn how to “code” by creating programs that move a turtle from a starting field to a diamond. Robot Turtles is reportedly the most successful game funded by the Kickstarter incubator. Players mus...
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Robot in a Maze Description: Filip (and his dad) started experimenting with robotics. They built a robot that moves through a maze drawn on a rectangular graph paper with some grid squares colored black. The robot’s color sensor prevents it from going on or through the black regions—they act like walls...
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Robotopia Description: In the wonderful land of Robotopia, life is better than ever. This is thanks to all the robots, who work hard together. Different types of robots come in different forms, that is, different numbers of arms and different numbers of legs. By day, when they are working, robots are g...
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Robots Description: While you weren’t watching, your $N$ robots have developed a life of their own and spread throughout your hometown. Each of your hometown’s $N$ intersections (numbered $0, \ldots , N-1$) contains exactly one robot. On each intersection $i$, there is exactly one red signpost ...
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Robots on a Grid Description: You have recently made a grid traversing robot that can find its way from the top left corner of a grid to the bottom right corner. However, you had forgotten all your AI programming skills, so you only programmed your robot to go rightwards and downwards (that’s after all...
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Rock Band Description: Every day after school, you and your friends get together and play in a band. Over the past couple of months, the band has been rehearsing a large number of songs. Now it’s time to go out and perform in front of a crowd for the first time. In order to do so, a set list for the co...
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Rock Climbing Description: Peter is attempting to deep-water solo a rock climbing cliff over the ocean. Deep-water soloing (DWS) is a form of solo rock climbing that relies solely upon the presence of water at the base of the climb to protect against injury from falling. Rock climbing is very exhausting a...
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Rock, Scissors, Paper Description: Bart’s sister Lisa has created a new civilization on a two-dimensional grid. At the outset each grid location may be occupied by one of three life forms: Rocks, Scissors, or Papers. Each day, differing life forms occupying horizontally or vertically adjacent grid loca...
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Rock-Paper-Scissors Tournament Description: Rock-Paper-Scissors is game for two players, $A$ and $B$, who each choose, independently of the other, one of rock, paper, or scissors. A player chosing paper wins over a player chosing rock; a player chosing scissors wins over a player chosing paper; a player ch...
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Rocket Powered Hovercraft Description: You are programming an autonomous rocket powered hovercraft. The vehicle can travel very, very fast, but turning is difficult. Since it’s hovering, it turns by firing directional thrusters, which will turn the vehicle whether it is moving forward or is stationary....
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Rocket Stages Description: Many rockets are made up of several stages to increase efficiency. When the fuel in one stage burns up, the stage can be discarded, reducing the weight of the remaining rocket. The first stage needs a strong engine capable of lifting the whole rocket, while later stages can h...
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Rocky Mountain Description: The Rocky Mountain Cable (RMC) company is planning to run cables from the top peak of the Rocky Mountains to lower points in the mountain range, so that cable cars can be used to transport tourists to the highest peak. A cable must connect from one of the potential sites to ...
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Roll Call Description: At the start of class, I like to call roll. I like to go through my list of students in alphabetical order. Where possible, I like to call students by their first names. Of course, if two students have the same first name, I have to also give the last name so they know who I’m ca...
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Roller Coaster Assembly Description: To make sure that the LTH campus maintains its long tradition of being the best campus in Sweden, several roller coasters will be constructed on and around campus. The roller coasters are constructed over an period of $N$ days, with one track segment being delivere...
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Roller Coaster Fun Description: Jimmy and his friends like to visit large theme parks. In the current theme park there are many roller coasters which then are categorized by Jimmy. He assigns a fun value to each coaster; however, the fun decreases with each run. More formally: for a specific roller coaste...
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Roman Holidays Description: The ancient Romans created many important things: aqueducts, really straight roads, togas, those candles that spout fireworks. But the most useless is Roman numerals, a very awkward way to represent positive integers. The Roman numeral system uses seven different letters, each ...
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Room Assignments Description: Once there was an inventor congress, where inventors from all over the world met in one place. The organizer of the congress reserved exactly one hotel room for each inventor. Each inventor, however, had its own preference regarding which room he would like to stay in. Bei...
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Room Evacuation Description: You are now the fire marshal. It is not a fun job to have. You have a layout of a room in the building as a 2D grid. There are known locations that people will occupy, there are known locations that people cannot walk into or out of, and there are known locations that are e...
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Room Painting Description: Joe’s landlord has allowed him to paint his room any colour that he wants, even multiple colours. Joe has come up with a very colourful design. Now he needs to buy the paint. Being a struggling student, Joe does not want to waste any money, so he has calculated the exact amou...
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Room Service Description: You are working for a company designing cute, funny robot vacuum cleaners. At a high level, the robots’ behavior is divided into three modes: * Exploration * Vacuuming * Rampant Killing Exploration Vacuuming Rampant Killing Unfortunately, while consumer testing shows that the l...
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Room to Grow Description: In an old orchard there is an alley of trees. Unfortunately, at the pace the trees are growing, the alley will become overgrown and the trees will start dying due to a lack of light and competition for nutrients. The $i$th tree in the alley is positioned $d_ i$ metres from the...
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Roomba 1 Description: Your friend Tómas has a very eccentric grandfather, mister Miyagi, from which he inherited a large shed in Raufarhöfn. The only condition his grandfather stipulated is that the floor must be kept clean. A rumor has long been going around regarding what’s stored in the shed, but so...
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Roomba 2 Description: Your friend Tómas has a very eccentric grandfather, mister Miyagi, from which he inherited a large shed in Raufarhöfn. The only condition his grandfather stipulated is that the floor must be kept clean. A rumor has long been going around regarding what’s stored in the shed, but so...
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Root of Minus One Description: You are given a prime $p$ satisfying $p = 1 \pmod{4}$ and should find a value $r$ that satisfies $0 \leq r < p$ and $r^2 = -1 \pmod{p}$. ## Input The first and only line of input contains a prime $p \leq 10^{18}$ satisfying $p = 1 \pmod{4}$. ## Output Print a ...
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Rooted Subtrees Description: A tree is a connected, acyclic, undirected graph with $n$ nodes and $n - 1$ edges. There is exactly one path between any pair of nodes. A rooted tree is a tree with a particular node selected as the root. Let $T$ be a tree and $T_ r$ be that tree rooted at node...
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Rot Description: Damir likes to rotate. Right now he is rotating tables of letters. He wrote an $R\times C$ table onto a piece of paper. He has also chosen an angle $K$, a multiple of $45$, and wants to rotate his table that many degrees clockwise. It turns out this task is a bit too hard for Dami...
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Rotate and Cut Description: Thomas is the best student in his class. One would think that it is fantastic for him to know all the stuff their teacher is speaking about... but unfortunately, he keeps becoming very very bored during the lessons. To prevent himself from falling asleep he draws silly drawi...
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Rotate to root Description: Rotate-to-root is a heuristic for balancing binary search trees. In this problem, the contents of the tree will be ignored, and you will be asked about the effect the heuristic has on the structure of the tree. A binary tree is either empty or consists of a node with a left ...
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Round Trips Description: Micah lives a peaceful life with his family in one of Canada’s most beautiful provinces. The vast wealth he has accumulated allows him to indulge in an extravagant hobby: collecting vintage automobiles. On Sunday afternoons, Micah and his family enjoy taking long, leisurely dri...
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Rounded Buttons Description: Most styles of graphical user interface make use of lots of rectangles. Buttons are rectangular, menus are rectangular and windows are rectangular. Some styles of user interface exhibit a softer appearance by using rounded rectangles for some components. As illustrated belo...
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Roundelays Description: Roundelay is a game where people dance in a circle holding hands, and some people might stand in the middle of the circle. You saw kids playing this game in two groups (two roundelays). Unfortunately, you completely forgot who was holding hands with whom. The only two things you...
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Routing Description: You work as an engineer for the Inane Collaboration for Performance Computing, where you are in charge of designing an intercommunication network for their computers. The network is arranged as a rectangular array of $2n - 1$ rows, each having $2^{n - 1}$ switches. A switch is ...
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Royal Delivery Description: Mario is submitting a letter via the royal mail to proclaim his love to Princess Peach. However, the Royal Toad Delivery Service is incredibly greedy and they charge for each character used in his message. However, Mario has devised a secret compression strategy known only t...
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Royal Family Tree Description: On September 9, 2015, Queen Elizabeth II became the longest reigning monarch in the history of the British Empire, surpassing the record set by her great-great-grandmother, Queen Victoria. This landmark event prompted many people to think about royal succession. Who is fi...
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Royal Routing Description: A king piece starts at A1 on a normal $8 \times 8$ chessboard. You are given a positive integer $n \leq 10^{18}$. How many ways are there for the king to make it to H8 in exactly $n$ steps? One step is one move for the king piece using normal chess rules. That is, one...
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Rubberboots Description: ## Input On the first line, the integer $n\in \{ 0,\ldots , 501\} $, specifying the number of instructions you have received. The follows a line with $n$ space-separated instructions. Each instruction is either an integer $k\in \{ 1,\ldots , 10^6\} $ meaning “walk forward ...
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Rubik's Rectangle Description: A new puzzle which aims to conquer the game market is a fusion of Rubik’s Cube and Fifteen. The board is an $H \times W$ frame with tiles with all numbers from $1$ to $H \cdot W$ printed on them. The only type of move that is allowed is flipping either one of the rows or...
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Rubik's Revenge in ... 2D!? 3D? Description: You are given a puzzle that can be represented as a $4 \times 4$ grid of colored cells. The solved puzzle contains 4 monochromatic rows, in this order: red, green, blue, yellow. Although we will analyze this puzzle using its 2D representation, it is actually...
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Ruffians Description: Ashley and Brandon are playing the new hit card game, Ruffians! In Ruffians, ten cards are dealt out in a grid of two rows and five columns. Each card has a number on it from 1 to 9. Ashley and Brandon are both looking for a pair of cards that have the same number. After playing...
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Ruglaður listi Description: Dagur likes taking walks downtown and one day when Dagur was on his daily walk, he bumped into an arcade machine in the middle of the street. Since Dagur is very curious, he just had to check this machine out. When he finally saw what was on the screen, he could not believe ...
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Rule N Description: A cellular automaton is an infinite one-dimensional array of cells. Each cell in the array has one of two states, $0$ or $1$, but the state of the cell may change over the course of an iterative process. At each iteration, each cell will take on a new state by applying a transit...
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Run-Length Encoding, Run! Description: Forrest lives in a prehistoric era of “dial-up Internet.” Unlike the fast streaming of today’s broadband era, dial-up connections are only capable of transmitting small amounts of text data at reasonable speeds. Forrest has noticed that his communications typicall...
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Running MoM Description: The bad thing about being an international Man of Mystery (MoM) is there’s usually someone who wants to kill you. Sometimes you have to stay on the run just to stay alive. You have to think ahead. You have to make sure you don’t end up trapped somewhere with no escape. Of cour...
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Running Race Description: You are the main organiser of an annual running race, where partipants must complete $k$ laps. The prizes are even larger than last year, and you have attracted more partipants than expected. Every participant has a unique start number for identification. In previous years, t...
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Running Routes Description: The administrators at Polygonal School want to increase enrollment, but they are unsure if their gym can support having more students. Unlike a normal, boring, rectangular gym, the gym floor at Polygonal is a regular $n$-sided polygon! They affectionately refer to the polygo...
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Running Steps Description: The coach wants his team members to run up the stadium steps taking either one or two steps with each stride so that: * The number of two step strides taken by each leg is the same. * The number of one step strides taken by each leg is the same. * The number of two ste...
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Russian Dolls Description: Russian nesting dolls are brightly painted hollow wooden figures. The dolls in a set have roughly the same shape, typically humanoid, but different sizes. When the set is assembled, the biggest doll contains the second-biggest doll, the second-biggest contains the third-bigge...
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Rust Description: Benni has just started playing the game Rust. This is an extremely exciting game full of action. But in Rust you can also build. Benni has been collecting rocks all day so he can build his own home. He has decided to build a square wall with dimensions $K \times K$. Benni never r...
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Ríkjafræði Description: Jörmunrekur spent his easter vacation reading up on a math subject known as category theory. It uses a lot of arrow diagrams to visualize the material. These diagrams display categories which consist of objects and arrows between objects. An important property of many of these d...
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Ríkjafræði 2 Description: Jörmunrekur thought he had achieved a decent understanding of category theory, but suddenly things got more complicated when he started looking deeper into things! Category theory often uses diagrams as visual aids for theorems or definitions. The diagrams usually represent ca...
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