Contents Maryland High School Programming Contest 1. 1 Stacked Floating Mountains 2. 2 Chess Puzzle 3. 3 Life Connections 4

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1 2010 Marylan High School Programming Contest 1 Contents 1 Stacke Floating Mountains 2 2 Chess Puzzle 3 3 Life Connections 4 4 Circle of Friens 5 5 Floating Mountain Stability 6 6 Aim It Right! 7 7 Navi Navigation 8 8 Locate Mining Center 10

2 2010 Marylan High School Programming Contest 2 1 Stacke Floating Mountains The floating mountains of Panora present a challenge for the human scientists, especially geologists an physicists, who have been trying to unerstan how such structures coul exist. While exploring the mountains, the scientists stumble across interesting stacke floating mountain structures, where ifferent mountains appeare stacke above one other, with the larger mountains being higher up in the stack. The scientists were able to calculate the size of each mountain, an they mae an interesting observation: that the sizes of the mountains forme a (generalize) Fibbonacci sequence. A sequence of numbers: x 1, x 2,..., x n, is calle a generalize Fibbonacci sequence if, for all i > 2, x i = x i 1 + x i 2 The stanar Fibbonacci sequence is simply a generalize Fibbonacci sequence with x 1 = x 2 = 1. An example of generalize Fibbonacci sequence is: 2, 5, 7, 12, 19,... Your goal is to help the scientists verify this conjecture. Specifically, you are to write a program that, given a sequence of numbers, ecies whether the sequence is a generalize Fibbonacci sequence or not. The first line in the test ata file contains the number of test cases, n. After that, each line contains one test case. The test case begins with the number of elements in the sequence, k, an then we have k numbers which form the sequence. Assume all numbers are 0, an that the numbers are all < For each test case, you are to output YES (if the sequence is a generalize Fibbonacci sequence) or NO (if it is not). 3 YES NO YES

3 2010 Marylan High School Programming Contest 3 2 Chess Puzzle Jake an Sully are playing aroun with a chessboar one night after working with their avatars all ay. They ecie it woul be interesting to place some rooks on the chessboar in a way that no rook can threaten another rook. Since rooks move along rows an columns, this means two rooks may not be on the same row or column. Your goal is to write a program to etermine whether any rooks are threatene. Chessboars are 8x8 boars with positions between (1,1) an (8,8). The input begins with the number of chess boars. Each chessboar is on a separate line an begins with the number of rooks, followe by the column an row positions of each rook. For each chessboar, your program shoul output the wors SAFE or NOT SAFE on a single line. 2 SAFE NOT SAFE

4 2010 Marylan High School Programming Contest 4 3 Life Connections On Panora all Navi are connecte by frienships. After carefully mapping frienships between ifferent Navi, Grace wants to measure the strength of the connection between pairs of Navi. She ecies the way to calculate this is to treat Navi as noes in a graph, an frienships between Navi as eges. Then the connection strength between two Navi can be efine as the number ifferent shortest paths each coul take to visit the other. Your goal is to help her calculate these values. Given a list of connections between Navi an two Navi u an v, you must compute the number of ifferent shortest paths between u an v. The length of the path is the number of Navi on the path. Two paths are ifferent if they pass through at least one ifferent Navi. Connections between Navi are escribe beginning with the line GRAPH BEGIN. Aitional lines lists iniviual Navi (noes), followe (on the same line) by their friens (eges). The line GRAPH END ens the list of connection escriptions. The next lines escribe pairs of Navi for which answers nee to be calculate, each on a single line. Following these lines, a new instance of the problem can be given, starting from scratch. You may assume all Navi are connecte (i.e., any Navi can reach another Navi by some path). Not all Navi will have their connections liste on a separate line: the frienships of some Navi may only be implie by the frienships given on other lines. Your output shoul consist of pairs of Navi in the same orer as in the input, followe by the number of shortest paths between them, both on a single line. For instance, in the following example the strength of the connection between Navi a an e is 2, since there are 2 paths of length 3 (the shortest possible) from a to e (a b e an a c e). Example: Example Example GRAPH BEGIN a b 1 a b c a c 1 b a 2 c a e 2 e b c 2 GRAPH END b e 1 a b a c a a e b c b e c a e b

5 2010 Marylan High School Programming Contest 5 4 Circle of Friens After measuring the strength of frienships between ifferent Navi, Grace wants to fin groups of Navi who form close-knit frienships. A group of friens has strength k if each Navi in the group has at least k friens within the group. Your goal is to help Grace fin the strongest, largest circle of friens for iniviual Navi. Connections between Navi are escribe beginning with the line GRAPH BEGIN. Aitional lines lists iniviual Navi, followe (on the same line) by their friens. The line GRAPH END ens the list of connection escriptions. The next lines escribe iniviual Navi to be analyze, each on a single line. Following these lines, a completely new instance of the problem can be given, starting from scratch. Some Navi may be only be liste as friens of other Navi (i.e., not all Navi will have their connections liste on a separate line). Your output shoul consist of one line for each Navi analyze, in the same orer they were liste in the input. Each line shoul contain the name of the Navi, the largest k for which the Navi is a member of some group of friens of strength k, an all the friens in that group (in alphabetical orer), incluing themselves. Every Navi in the group must know the initial Navi either irectly or inirectly through some sequence of common friens (i.e., the frienship graph must be connecte). In the example below, Navi c is a member of a group of friens of strength 3: bce. She is also a member of several groups of friens of strength 2 (bc, bce, ce,... ) but because 3 > 2, the group of strength 3 is the one that shoul be output. Navi f is a member of several groups of strength 1 (ef, bef, ef,... ) but the largest one is abcef, so that is the one that shoul be output. Example: GRAPH BEGIN a b c b c e c e e f GRAPH END a b c e f a 2 a b c e b 3 b c e c 3 b c e 3 b c e e 3 b c e f 1 a b c e f c a b e c 3 b c e f

6 2010 Marylan High School Programming Contest 6 5 Floating Mountain Stability After receiving your program from Problem 1, the scientists use it to try to verify their conjecture (that the sizes of stacke floating mountains forme a generalize Fibbonacci sequence). However, although they were able to verify the conjecture for a large number of cases, they iscovere that there were stacke structures that i not satisfy the property. Further, they also iscovere that floating mountains may have negative weights (they conjecture that this has to o with some unique properties of Unobtainium ). The scientists now believe that the sizes of the stacke mountains i follow generalize Fibbonacci sequence property originally (when they were forme), but they believe that some of the mountains in the structures may have been estroye or may have rifte apart. They further observe that at most 9 consecutive mountains in the stack may be remove without compromising the stability of the structure. They are now trying to verify this new conjecture. You are to write a program for this purpose. Specifically, given a sequence of numbers, some of which may be negative, you must etermine if the numbers are part of a generalize Fibbonacci sequence (let s call it the original sequence), such that all consecutive pairs of numbers in the input sequence are less than 10 apart (i.e., fewer than 9 items between any consecutive pair of numbers) in the original generalize Fibbonacci sequence. As an example, the sequence: , follows this property because the numbers are from the following generalize Fibbonacci sequence: an 0 & 6 are only 4 numbers apart in the generalize sequence. As another example: the sequence , also satisfies the property. Here is the corresponing generalize Fibbonacci sequence: The first line in the test ata file contains the number of test cases, n. After that, each line contains one test case. The test case begins with the number of elements in the sequence, k (k < 50), an then we have k numbers which form the sequence. Assume the numbers are all > 2 30 an < For each test case, you are to output STABLE (if the sequence satisfies the property) followe by the first five elements of the generalize Fibbonacci sequence (beginning with the first number in the input sequence), or UNSTABLE (if it oes not). If multiple generalize Fibbonacci sequences are possible, select the sequence with the smallest gap (i.e., number of missing numbers) between the first two numbers. 3 STABLE STABLE UNSTABLE

7 2010 Marylan High School Programming Contest 7 6 Aim It Right! There are many similarities between the human an the Navi cultures an lifestyles. For instance, their sports an boargames bear similarities to ifferent sports an games on Earth. But at the same time, there are significant ifferences (e.g., instea of water polo, they play air polo on their Banshees). There is also a version of Billiars, which is quite similar to our version, but with the ifference that their Billiars table has a large circular hole right at the center (see figure). Of course, a person who hits the ball into that hole immeiately loses. Jake an Tsu tey start a frienly game of Billiars which soon turns into a not-so-frienly game with possibly the leaership of the Omaticaya at stake. In each roun, they take turns. At each turn, an impartial juge places the cue ball at one location on the table (call it point A) an another ball somewhere else (call it point B). The person whose turn it is, must hit the ball at point B from point A with minimum number of rebouns off the walls of the table (see figure). Your goal is to help Jake win the game an retain his leaership (otherwise he will have to try to tame Toruk once again, an only so many times you can survive that). Given the input parameters, you are to fin the minimum number of rebouns neee to get from point A to point B (or eclare that it is not possible to o so). Assume the boar is perfectly frictionless an the walls are infinitely elastic. So a ball hit at an angle θ 1 will reboun in the opposite irection at exactly the same angle. The Billiars table is a square with imensions 100 by 100. Assume that the center of the table (an hence the center of the hole at the mile) is at (0, 0). You are given the x an y coorinates of the points A an B, an also the raius of the hole in the mile (r). The first line of the input file is the number of test cases. Each line contains 5 integers: A x, A y, B x, B y, r, where (A x, A y ) are the coorinates of point A, an (B x, B y ) are the coorinates of point B. The value r enotes the raius. You can assume that neither point A nor point B are in the hole, an that r < 50. For each test case, you are to fin the minimum number of rebouns neee to reach from point A to point B. If it is less than 10, output on a single line REBOUNDS followe by the number of rebouns neee. If it is not possible to hit point B from point A in < 10 rebouns, output on a single line NOT POSSIBLE. 2 REBOUNDS NOT POSSIBLE θ 1 θ 1 A B

8 2010 Marylan High School Programming Contest 8 7 Navi Navigation The Navi villages on Panora are part of gigantic hometrees. Hometrees specialize in proucing ifferent types of fruits that Navi like to eat. Neytiri s mother Mo at asks her to calculate the shortest path between two given hometrees that allows Mo at to collect every ifferent type of fruit exactly once. Your goal is to help Neytiri calculate these paths. Warning: since there are many hometrees on Panora, you will not be able to simply examine all possible paths an select the least expensive. Paths between hometrees are escribe beginning with the line GRAPH BEGIN. Aitional lines lists iniviual hometrees (noes), the type of fruit prouce by the hometree, the istance to the neighboring hometrees, followe (on the same line) by their neighboring hometrees (eges). The line GRAPH END ens the list of connection escriptions. The next lines escribe pairs of hometrees for which answers nee to be calculate, each on a single line. Following these lines, a completely new instance of the problem can be given, starting from scratch. You may assume any hometree can reach any other hometree by some path. Each hometree will be liste at least once as the first item on some line between the GRAPH BEGIN an GRAPH END. The same hometree can be liste more than once with ifferent istance values, but it must always have the same type of fruit assigne to it. Iniviual connections can appear at most once. It is vali to list only a hometree an its color (specifying no new connections). Fruit names will be integers. Not all integers have to be use, however. Your path nee only try to collect fruits that at least one tree grows. Your output shoul consist of pairs of hometrees in the same orer as in the input, followe by the length of the shortest path between them that collects each type of fruit exactly once. If such a path oes not exist, you shoul output NONE. In the first example below, the path a b c collects all the fruit types (1, 2, 3, an 5) an the path has length 4.0. No goo path exists between a an c, however: the path a b c c woul collect all the fruit types, but it collects fruit 1 twice!

9 2010 Marylan High School Programming Contest 9 GRAPH BEGIN a 4.0 a 3 1 b e a c NONE b 2 2 c h e 6.0 c e 2 GRAPH END a a c GRAPH BEGIN e 1 2 f e 1 3 g f 2 g 2 h 3 4 g f GRAPH END h e e 1 2 a 1 b 2 c Shae boxes give the type of fruit grown at each noe.

10 2010 Marylan High School Programming Contest 10 8 Locate Mining Center As part of a peace treaty with the Navi, the humans are allowe mine for Unobtainium in a remote, eserte area of Panora. The scientists have ientifie many possible excavation sites, an are now trying to figure out where to place the Mining Center. The placement of the mining center is further constraine by the fact that the robots that will carry the Unobtainium from the excavation sites to the mining center can only walk along prespecifie Gri lines (see figure). The goal is now to fin the location of the mining center so that the maximum istance to the excavation sites is minimize. You are to write a program for fining that location. Specifically, you are given the x- an y-coorinates of the excavations sites, (x 1, y 1 ), (x 2, y 2 ),...(x n, y n ) (n enotes the number of excavation sites). You are to fin the coorinates for the Mining Center, (x 0, y 0 ), so that the maximum of the istances between the mining center an the excavation sites is minimize. All coorinates must be integers. The istance metric to be use is the rectilinear istance (also calle Manhattan istance). Specifically, the rectilinear istance between (x 0, y 0 ) an (x i, y i ) is: x i x 0 + y i y 0 Further, if there are multiple locations which qualify, then you must fin the location that is closest to the origin by Eucliean istance metric. Since origin is at (0, 0), this correspons to minimizing x y2 0. In other wors, given the input (x 1, y 1 ),..., (x n, y n ), your goal is to fin (x 0, y 0 ) such that: max( x i x 0 + y i y 0 ) i is minimize, an if there are multiple answers, then x y2 0 is the smallest among all such answers.

11 2010 Marylan High School Programming Contest 11 The first line in the test ata file contains the number of test cases, n. After that, each line contains one test case. The test case begins with the number of mines, followe by the x an y coorinates of each mine. All coorinates must be integers. Very large coorinate values may be use (million+), so brute force methos will not work. For each test case, you are to output a line beginning with LOCATION, followe by the x an y coorinates of the mining center. All coorinates must be integers. 2 LOCATION LOCATION

12 2010 Marylan High School Programming Contest 12 Practice 1 Checkerboar Rows Colonel Quaritch is playing checkers one ay, an ecies it woul be interesting to write a program to calculate the maximum number of pieces on a single row. Checkerboars are 8x8 boars with positions between (1,1) an (8,8). The input begins with the number of boars. Each boar is on a separate line an begins with the number of pieces, followe by the column an row positions of each piece. For each checkerboar, your program shoul output the maximum number pieces on any one row

13 2010 Marylan High School Programming Contest 13 Practice 2 Mining Maps Aministrator Selfrige is analyzing possible mining routes on Panora. He has collecte some ata in the form of a graph. Your goal is to help him collect some information about each graph. Connections between mines are escribe beginning with the line GRAPH BEGIN. Aitional lines lists iniviual mines (noes), followe (on the same line) by their neighboring mines (eges). The line GRAPH END ens the list of path escriptions. The entire problem may be repeate as esire, starting from scratch each time. Some mines may appear only as neighboring mines, without being escribe on a separate line. Your output shoul consist one line (for each graph analyze), consisting of NODES followe by the number of noes, followe by EDGES an the number of eges in the graph. GRAPH BEGIN NODES 3 EDGES 2 a b NODES 6 EDGES 7 b c GRAPH END GRAPH BEGIN a b c b c e f e f GRAPH END e f b a c

14 2010 Marylan High School Programming Contest 14 Practice 3 Hauling Ore Aministrator Selfrige is analyzing possible mining routes on Panora. He has collecte some ata in the form of a graph. The latest ore carriers can visit exactly 3 mining camps. Your goal is to help him fin out which mines may be visite from other mines with 2 stops (but not fewer). Connections between mines are escribe beginning with the line GRAPH BEGIN. Aitional lines lists iniviual mines (noes), followe (on the same line) by their neighboring mines (eges). The line GRAPH END ens the list of path escriptions. The next lines list mines for which answers nee to be calculate, each on a single line. Following these lines, a completely new instance of the problem can be given, starting from scratch. Some mines may appear only as neighboring mines, without being escribe on a separate line. Mine names can be arbitrary strings, as long as they on t contain any whitespace. Your output shoul consist one line (for each mine analyze), consisting of the name of the mine, followe by the mines, in alphabetical orer, that can be visite with exactly 2 stops but not fewer, starting from the given mine. GRAPH BEGIN a b c b c e f GRAPH END a b c e f e f a c b f b e e f b a c