Jolly Jumpers..~

Problem Statement
     
You're given a set of inequalities. Each of the inequalities refers to the variable X. Determine the maximum subset of the given set which has a solution.  To make your task easier, the inequalities in the given set are always reduced to one of the following five forms:
 X < C
 X <= C
 X = C
 X > C
 X >= C
Here, C indicates some non-negative integer constant.  The inequalities are given in the vector <string> inequalities, where each element is a single inequality formatted as shown above. Return the maximal number of inequalities of the set which can be satisfied simultaneously.
Definition
     
Class:
Inequalities
Method:
maximumSubset
Parameters:
vector <string>
Returns:
int
Method signature:
int maximumSubset(vector <string> inequalities)
(be sure your method is public)
    
 
Notes
-
Note that X doesn't have to be an integer or positive number.
Constraints
-
inequalities will contain between 1 and 50 elements, inclusive.
-
Each element of inequalities will be formatted "X <E> <C>", where 'X' is uppercase, <E> is one of "<", "<=", "=", ">=" or ">", and <C> is an integer between 0 and 1000, inclusive, with no extra leading zeroes (all quotes for clarity).
-
No two elements of inequalities will be equal.
Examples
0)
 
    
{"X <= 12","X = 13","X > 9","X < 10","X >= 14"}
Returns: 3
Any value between 9 and 10 will satisfy the first, third and fourth inequalities.
1)
 
    
{"X < 0","X <= 0"}
Returns: 2
The solution to the whole set is any negative number.
2)
 
    
{"X = 1","X = 2","X = 3","X > 0"}
Returns: 2
Obviously, you can choose no more than one equality in addition to the fourth inequality.
3)
 
    
{"X <= 521","X >= 521","X = 521","X > 902","X > 12","X <= 1000"}
Returns: 5
The best choice is number 521.
This problem statement is the exclusive and proprietary property of TopCoder, Inc. Any unauthorized use or reproduction of this information without the prior written consent of TopCoder, Inc. is strictly prohibited. (c)2003, TopCoder, Inc. All rights reserved.

Problem Statement
     
Suppose that there are N numbers written in a row. A row above this one consists of N-1 numbers, the i-th of which is the sum of the i-th and (i+1)-th elements of the first row. Every next row contains one number less than the previous one and every element is the sum of the two corresponding elements in the row below. The N-th row contains a single number. For example, if the initial numbers are {2,1,2,4}, the whole structure will look like this:
    15
   6 9
  3 3 6
 2 1 2 4
We shall refer to such a structure as a number pyramid. Two number pyramids are equal if all the numbers at corresponding positions are equal.  Given ints baseLength and top, compute the total number of different number pyramids consisting of positive integers, having baseLength elements in the first row and the value at the top equal to top. Since the number of such pyramids might be enormous, return the result modulo 1,000,000,009.
Definition
     
Class:
NumberPyramids
Method:
count
Parameters:
int, int
Returns:
int
Method signature:
int count(int baseLength, int top)
(be sure your method is public)
    
 
Constraints
-
baseLength will be between 2 and 1,000,000, inclusive.
-
top will be between 1 and 1,000,000, inclusive.
Examples
0)
 
    
3
5
Returns: 2
The following are two possible pyramids with 3 numbers in the base and the number 5 at the top: 
1)
 
    
5
16
Returns: 1
The only number pyramid with base of size 5 and 16 at the top looks like this: 
2)
 
    
4
15
Returns: 24
 
3)
 
    
15
31556
Returns: 74280915
 
4)
 
    
150
500
Returns: 0
 
This problem statement is the exclusive and proprietary property of TopCoder, Inc. Any unauthorized use or reproduction of this information without the prior written consent of TopCoder, Inc. is strictly prohibited. (c)2003, TopCoder, Inc. All rights reserved.

Problem Statement
     
A contest is held between two teams of fighters. The first team is comprised of N persons and the second is comprised of M persons. In the course of the contest, each fighter of the first team will confront each fighter of the second team once. To make the contest more entertaining and less protracted, the organizers divided it into several rounds. In each of the rounds, several fights occur simultaneously. Therefore, a fighter may only participate in one fight per round.  The contest must be divided into the least number of rounds. One can easily see that this number is the maximum between N and M. Now the exact schedule of fights must be determined. Return a vector <string> containing exactly N elements. Each element of the return must contain exactly M characters from the set {'1'-'9', 'A'-'Z', 'a'-'o'}. Letters 'A'-'Z' stand for numbers 10-35 and 'a'-'o' stand for numbers 36-50. The j-th character of element i corresponds to the number of the round in which fighter i of the first team encounters fighter j of the second team. If there are several possible schedules, return the one with the lexicographically least first element. If there still are multiple choices, choose the schedule with the lexicographically least second element, and so on.
Definition
    
Class:
TheContest
Method:
getSchedule
Parameters:
int, int
Returns:
vector <string>
Method signature:
vector <string> getSchedule(int N, int M)
(be sure your method is public)
    
 
Notes
-
A string A is lexicographically less than another string B of the same length if there exists a position i such that each character of A before the i-th position is equal to the character at the corresponding position in B, and A[i] is less than B[i]. When comparing the characters, refer to the following list of characters in ascending order: '1', '2', ..., '9', 'A', 'B', ..., 'Z', 'a', 'b', ..., 'o'.
Constraints
-
N will be between 1 and 50, inclusive.
-
M will be between 1 and 50, inclusive.
Examples
0)
 
    
3
3
Returns: {"123", "231", "312" }
There are three persons on each team, so the whole contest is divided into three rounds. If we denote the i-th person of the first team as Ai and the j-th person of the second team as Bj, the fights schedule for each of the rounds is: Round 1: A1-B1, A2-B3, A3-B2 Round 2: A1-B2, A2-B1, A3-B3 Round 3: A1-B3, A2-B2, A3-B1
1)
 
    
4
4
Returns: {"1234", "2143", "3412", "4321" }
This time, both teams comprise of four persons, so the contest takes 4 rounds to finish. Round 1: A1-B1, A2-B2, A3-B3, A4-B4 Round 2: A1-B2, A2-B1, A3-B4, A4-B3 Round 3: A1-B3, A2-B4, A3-B1, A4-B2 Round 4: A1-B4, A2-B3, A3-B2, A4-B1
2)
 
    
4
6
Returns: {"123456", "214365", "345612", "436521" }
The competition now is held in 6 rounds. Note that since the teams are of unequal size, two of the second team's fighters are taking rest in each of the rounds. Round 1: A1-B1, A2-B2, A3-B5, A4-B6 Round 2: A1-B2, A2-B1, A3-B6, A4-B5 Round 3: A1-B3, A2-B4, A3-B1, A4-B2 Round 4: A1-B4, A2-B3, A3-B2, A4-B1 Round 5: A1-B5, A2-B6, A3-B3, A4-B4 Round 6: A1-B6, A2-B5, A3-B4, A4-B3
3)
 
    
5
3
Returns: {"123", "214", "345", "451", "532" }
 
4)
 
    
28
40
Returns:
{"123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcde",
 "21436587A9CBEDGFIHKJMLONQPSRUTWVYXaZcbed",
 "34127856BC9AFGDEJKHINOLMRSPQVWTUZaXYdebc",
 "43218765CBA9GFEDKJIHONMLSRQPWVUTaZYXedcb",
 "56781234DEFG9ABCLMNOHIJKTUVWPQRSbcdeXYZa",
 "65872143EDGFA9CBMLONIHKJUTWVQPSRcbedYXaZ",
 "78563412FGDEBC9ANOLMJKHIVWTURSPQdebcZaXY",
 "87654321GFEDCBA9ONMLKJIHWVUTSRQPedcbaZYX",
 "9ABCDEFG12345678PQRSTUVWXYZabcdeHIJKLMNO",
 "A9CBEDGF21436587QPSRUTWVYXaZcbedIHKJMLON",
 "BC9AFGDE34127856RSPQVWTUZaXYdebcJKHINOLM",
 "CBA9GFED43218765SRQPWVUTaZYXedcbKJIHONML",
 "DEFG9ABC56781234TUVWPQRSbcdeXYZaLMNOHIJK",
 "EDGFA9CB65872143UTWVQPSRcbedYXaZMLONIHKJ",
 "FGDEBC9A78563412VWTURSPQdebcZaXYNOLMJKHI",
 "GFEDCBA987654321WVUTSRQPedcbaZYXONMLKJIH",
 "HIJKLMNOPQRSTUVWXYZabcde123456789ABCDEFG",
 "IHKJMLONQPSRUTWVYXaZcbed21436587A9CBEDGF",
 "JKHINOLMRSPQVWTUZaXYdebc34127856BC9AFGDE",
 "KJIHONMLSRQPWVUTaZYXedcb43218765CBA9GFED",
 "LMNOHIJKTUVWPQRSbcdeXYZa56781234DEFG9ABC",
 "MLONIHKJUTWVQPSRcbedYXaZ65872143EDGFA9CB",
 "NOLMJKHIVWTURSPQdebcZaXY78563412FGDEBC9A",
 "ONMLKJIHWVUTSRQPedcbaZYX87654321GFEDCBA9",
 "PQRSTUVWXYZabcde9ABCDEFGHIJKLMNO12345678",
 "QPSRUTWVYXaZcbedA9CBEDGFIHKJMLON21436587",
 "RSPQVWTUZaXYdebcBC9AFGDEJKHINOLM34127856",
 "SRQPWVUTaZYXedcbCBA9GFEDKJIHONML43218765" }
 
This problem statement is the exclusive and proprietary property of TopCoder, Inc. Any unauthorized use or reproduction of this information without the prior written consent of TopCoder, Inc. is strictly prohibited. (c)2003, TopCoder, Inc. All rights reserved.