FIU Team Qualifier Competition

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Garry Owens
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1 FIU Team Qualifier Competition Problem Set Jan 22, 2016 A: Deck of Cards B: Digit Permutation C: Exchanging Letters D: Iconian Symbols E: Mines of Rigel F: Snowman s Hat G: Robby Explores Mars

2 A: Deck of Cards Suppose we start with a deck of n unique cards and deal the entire deck out to k players in the usual way: the top card is given to player 1, the next is given to player 2, the k th card is given to player k, the k+1 st to player 1, and so on. Next, we will pick up the cards, by placing player 1 s cards on top, followed by player 2's cards, and so on, until player k s cards are on the bottom. Also, each player s cards are in reverse order the last card they were dealt is on top, and the first card they were dealt is on the bottom. How many times, including the first, must this process be repeated before the deck returns to its original order? The The input file will contain a maximum of 100 test cases. Each test case will consist of a single line with two integers, n and k (1 n 800, 1 k 800). The input will end with a line containing two 0 s. The Output For each test case in the input, print a single integer, indicating the number of deals required to return the deck to its original order. Output each integer on its own line, with no extra spaces, and no blank lines between answers. All possible inputs yield answers which will fit in a signed 64-bit integer

3 B: Digit permutation Given a base-16 integer with N digits, how many unique integers, with N digits and no leading zeros, can you generate by re-arranging its digits. For example, for integer18f there are 6 unique integers: 18F 1F8 81F 8F1 F18 F81 Constraints : 1 n 19 begins with T, indicating the number of test cases. On the following T lines, each test case begins with an integer N followed by and base-16 integer with N digits. Output For each test case, output a single integer with the answer F B FCA

4 C: Exchanging Letters Suppose you are given a list of words, all in uppercase letters with no spaces, and you want to mix them up by exchanging letters in the words. Let s assume there are n words, each containing m letters. We will number the words starting with index 1, in the order they appear in the input data. For example, let s start with integers i, j, k (1 i < j n, 1 k m); we can select words indexed i and j, and swap the prefixes of length k between the two words. For example, using the list (AAB, BAA) and k=1, the new list becomes (BAB, AAA), and BAB is in position 1. We can change (BAB,AAA) to (AAA,BAB) if k=3. Also, we can change (AAA, BAB) to (BAA, AAB) if k=2. Altogether, 4 words have appeared in index position 1: AAB, BAB, AAA, and BAA. Your task is to figure out how many different words can appear in position 1 if you perform any number of swaps. As you perform each swap, you can freely choose values for i, j, k independently from the previous swap. The resulting number of different words can be very large, so you must reduce it to a value modulo ( ). The first integer T indicates the number of test cases to follow (1 T 50). Then each test case begins with two integers n and m (1 n, m 100) the number of names, followed by the length of each word. The following n lines contain words, where each word contains exactly m uppercase letters (A..Z). Output Print a single integer indicating the number of different words that could end up in position number 1 in the list after the applying all swaps according to our rules. Print the number modulo ( ) AAB BAA 2 5 CXDAM AGBRD 4 32

5 D: Iconian Symbols You are helping Professor P, an expert archaeologist, decipher symbols left behind by the Iconians, an ancient civilization. You may recall that the Enterprise found their planet in the Romulan neutral zone. Professor P knows this ancient society used a Base 10 system, and that they never started a number with a leading zero unless it is exactly zero (single digit). He figured out most of the digits, as well as a few operators, but he needs your help to figure out the rest. The professor will give you a simple math expression. He has converted all of the symbols he knows into digits. The only operators he knows are addition (+), subtraction (-), and multiplication (*), so only these will appear. Each integer will be in the range from to , and will consist of only the digits 0-9, possibly a leading -, and possibly a few?s. The?s represent a digit symbol the professor doesn't know (never an operator, =, or leading minus sign). All of the?s in an expression will represent the same digit (0-9), and it will not be one of the other given digits in the expression. Given an expression, figure out the value of the symbol represented by the question mark. If more than one digit works, give the lowest one. If no digit works, that indicates bad news for the professor -- it means some of his symbols were not copied correctly. Output negative 1 in that case. The first line will contain the integer T (1 <= T <= 100), indicating the number of test cases to follow. Following this line will be T test cases, each on a separate line, in the following format: [number][op][number]=[number] Each [number] will consist of only the digits 0-9, with possibly a single leading minus (-), and possibly some?s. No number will begin with a leading 0 unless it is 0, no number will begin with -0, and no number will have more than 6 places (digits or?s). The [op] will separate the first and second [number]s, and will be one of: +, - or *. The = sign will always be present between the second and third [number]s. There will be no spaces, tabs, or other characters. There is guaranteed to be at least one? (question mark) in every equation. Output Output a single line with the lowest digit that will make the equation work when substituted for the?s, or output -1 if no digit will work. Do not output any spaces =? 123*45?=5?088-5?*-1=5? =5???*??=302?

6 E: Mines of Rigel The Rigel mining consortium wants to drill a series of underground tunnels to connect each of the mines within a few square miles of Rigel III. As you know, the climate on this world is very severe, consisting of methane and other toxic gases. They have a list of mine entrances (identified by X/Y point coordinates) that must all be connected, but there are some limitations. The tunneling equipment can only move in straight lines between points. Tunnels cannot cross each other, but any number of tunnels can meet at the same mine entrance. Given a list of mine entrances, what is the smallest overall tunnel length necessary to make sure that every pair of mine entrances is connected, either directly, or indirectly? There will be fewer than 100 test cases in the input. Each test case will begin with an integer N (2 N 1,000), which is the number of mine entrances. On each of the next N lines will be two integers, X and Y ( 1,000 X,Y 1,000), which are the (X,Y) locations of the N entrances. Within a test case, all entrance locations will be distinct. The input will end with a line containing a single 0. Output For each test case, output a single real number, representing the least total tunnel lengths the Rigelian miners will need to connect all the mines. Print this number with exactly two decimal places, rounded. Print each number on its own line with no spaces. Do not print any blank lines between answers

7 F: Snowman s Hat Suppose you want to draw a two-dimensional snowman with a perfectly round head with radius R. After doing this, you might want to draw large hat on him, in the shape of a 2-D isosceles triangle, where the base is the side with a different length. But the hat cannot be too tall, otherwise the wind will blow it off. Therefore, you have asked your friend to tell you what the maximum height of the hat should be, starting at the top of the head. He comes up with H. The hat will have a height of H from the top of the snowman s head, but the total height of the hat will be greater than H. We would like you to calculate the largest possible area the hat could have, given the values of R and H. Constraints: The base of the hat must be shorter than the diameter of the head, and the two legs of the isoceles triangle must touch the circle. The ranges of R and H are: 0 < R < , and 0 < H < On the right is an example of a valid hat. : The first line contains T, the number of test cases to follow. Each test case contains two real numbers R, H, on a separate line. Output: For each test case, output one real number showing the largest possible area of the hat. Round your answer to the nearest one-hundredth. : :

8 G: Robbie Explores Mars Robbie the robot has been sent to explore the planet Mars. Each morning, the base station must send Robbie a navigation program that will move him from his current location to some destination. A navigation program consists of a sequence containing the following three commands, repeated in any order: FORWARD K: move forward by K units. TURN LEFT: turn left (in place) by 90 degrees. TURN RIGHT: turn right (in place) by 90 degrees. Robbie also has sensor units which allow him to obtain a map of his surrounding area, represented as a grid. Some grid points contain obstructions such as craters, so the program must avoid these points or risk losing the robot. If Robbie s initial location, the direction he is facing, and his destination position are known, you must create the shortest possible program (consisting of the fewest commands) to move Robbie to his destination (we do not care which direction he faces at the destination). You must calculate the number of different shortest programs that can move Robbie to his destination. However, the number of shortest programs can be very large, so you are satisfied to compute the number as a remainder modulo 1,000,000. There will be several test cases in the input. Each test case will begin with a line with two integers N M, where N is the number of rows in the grid, and M is the number of columns in the grid (2 N, M 100). The next N lines of input will have M characters each. The characters will be one of the following:. Indicates a navigable grid point. * Indicates a crater (i.e. a non-navigable grid point). X Indicates the target grid point. There will be exactly one X. N, E, S, or W Indicates the starting point and initial heading of the robot. There will be exactly one of these. The directions mirror compass directions on a map: N is North (toward the top of the grid), E is East (toward the right of the grid), S is South (toward the bottom of the grid) and W is West (toward the left of the grid). There will be no spaces and no other characters in the description of the map. The input will end with a line containing two 0 s.

9 Output For each test case, output two integers on a single line, with a single space between them. The first is the length of a shortest possible program to navigate Robbie from his starting point to the target, and the second is the number of different programs of that length which will get Robbie to the target (modulo 1,000,000). If there is no path from Robbie to the target, output two zeros separated by a single space. Output no extra spaces, and do not separate answers with blank lines. 5 6 *...X...*...*...* N...* X.****.****.****.**** N**** 3 3.E. ***.X

UNC Charlotte 2002 Comprehensive. March 4, 2002

UNC Charlotte 2002 Comprehensive. March 4, 2002 UNC Charlotte March 4, 2002 1 It takes 852 digits to number the pages of a book consecutively How many pages are there in the book? A) 184 B) 235 C) 320 D) 368 E) 425 2 Solve the equation 8 1 6 + x 1 3