Jolly Jumpers..~

Problem Statement
     
A Ferrers diagram of the partition of positive number n = a1 + a2 + ... + ak, for a list a1, a2, ..., ak of k positive integers with a1 ≥ a2 ≥ ... ≥ ak is an arrangement of n boxes in k rows, such that the boxes are left-justified, the first row is of length a1, the second row is of length a2, and so on, with the kth row of length ak. Let's call a FIELD diagram of order fieldOrder a Ferrers diagram with a1 ≤ fieldOrder, a2 ≤ fieldOrder - 1, ..., ak ≤ fieldOrder - k + 1, so a FIELD diagram can have a number of rows which is less than or equal to fieldOrder. Your method will be given fieldOrder, it should return the total number of FIELD diagrams of order fieldOrder.
Definition
   
Class:
FIELDDiagrams
Method:
countDiagrams
Parameters:
int
Returns:
long long
Method signature:
long long countDiagrams(int fieldOrder)
(be sure your method is public)
   
 
Constraints
-
fieldOrder will be between 1 and 30, inclusive
Examples
0)
 
   
2
Returns: 4

Problem Statement
     
Particles (which can be considered points in 3D-space for the purposes of the problem) can move in an electro-magnetic field. If a particle is charged, its trajectory can be described as spiral, and if it is uncharged, it is just a straight line. Given two particles (one charged and one uncharged) it should be determined whether they can possibly collide or not. Two particles can possibly collide if and only if their trajectories intersect.
Some steps have already been made by the physicist to simplify the problem, so the coordinates of the charged particle are represented as follows:
x1 = cos(PI * t)
y1 = sin(PI * t)
z1 = t
and for the uncharged particle:
x2 = vx * t + x0
y2 = vy * t + y0
z2 = vz * t + z0
Here t is a parameter which can be chosen arbitrarily and independently for both trajectories.
Your method will be given 6 integers - vx, vy, vz, x0, y0 and z0, describing the trajectory of the uncharged particle. It should determine whether the two given trajectories intersect or not. If they do, it should return a vector <double> containing exactly 3 elements x, y and z - the coordinates of the point where a collision can happen. If there is more than one such point, it should return a vector <double> containing exactly three zeroes. If collision of the two particles is impossible it should return an empty vector <double>.
Definition
   
Class:
ParticleCollision
Method:
collision
Parameters:
int, int, int, int, int, int
Returns:
vector <double>
Method signature:
vector <double> collision(int vx, int vy, int vz, int x0, int y0, int z0)
(be sure your method is public)
   
 
Notes
-
PI can be considered equal to 3.14159265358979323846.
-
All return values with either an absolute or relative error of less than 1.0E-9 are considered correct.
Constraints
-
vx, vy and vz will each be between -10 and 10, inclusive.
-
x0, y0 and z0 will each be between -10 and 10, inclusive.
Examples
0)
 
   
0
0
0
0
0
0
Returns: { }
The second trajectory is a single point (0, 0, 0), which doesn't lie on the first trajectory.
1)
 
   
2
4
1
-1
-1
0
Returns: {0.0, 1.0, 0.5 }
There is a single intersection point with coordinates (0, 1, 0.5).
2)
 
   
4
4
2
5
4
0
Returns: {0.0, 0.0, 0.0 }
There are two intersection points.
3)
 
   
0
0
1
1
0
0
Returns: {0.0, 0.0, 0.0 }
There are infinitely many intersection points.
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
     
You are given a convex polygon and you must count the number of N-cool points. A point is N-cool if it is a lattice point covered by the polygon and also an endpoint of some N-cool segment. A line segment is N-cool if it contains at least N lattice points that are covered by the polygon. A point is considered covered by the polygon if it is inside or on the boundary of the polygon. Endpoints of a segment are considered to be inside of it. A lattice point is a point with integer coordinates.
Consider the example in the picture below. Here N is equal to 6, and there are 21 6-cool points, marked green. Also two 6-cool segments, which are colored red, are shown.
 
You are given vector <int>s x and y describing the vertices of the polygon (in no particular order). The coordinates of each vertex are specified by corresponding elements of x and y. Return the number of N-cool points.
Definition
   
Class:
NCool
Method:
nCoolPoints
Parameters:
vector <int>, vector <int>, int
Returns:
int
Method signature:
int nCoolPoints(vector <int> x, vector <int> y, int n)
(be sure your method is public)
   
 
Notes
-
A polygon is convex if it contains all the line segments connecting any pair of its points.
Constraints
-
x and y will each contain between 3 and 50 elements, inclusive.
-
x and y will contain the same number of elements.
-
All elements of x and y will be between 0 and 10000, inclusive.
-
The number of lattice points inside or on the boundary of the specified polygon will not be greater than 500000.
-
N will be between 2 and 500000, inclusive.
-
The polygon specified by x and y will be convex and will have nonzero area.
Examples
0)
 
   
{0, 1, 2, 7, 7}
{3, 1, 6, 1, 5}
6
Returns: 21
This is the example from the problem statement.
1)
 
   
{0, 1, 0}
{0, 0, 1}
2
Returns: 3
These three points form a triangle, whose sides are 2-cool segments, so all three vertices are 2-cool.
2)
 
   
{0, 0, 1, 2, 2, 1, 0, 0, 2}
{0, 1, 2, 2, 1, 0, 0, 0, 2}
3
Returns: 6
One point can be mentioned in the input two or more times.
3)
 
   
{0, 1, 1, 2, 2, 3, 3, 4, 4, 5}
{1, 0, 2, 0, 2, 0, 2, 0, 2, 1}
5
Returns: 4
There can be 3 points of the polygon, lying on the same line.
4)
 
   
{0, 1, 1, 2, 2, 3, 3, 4, 4, 5}
{1, 0, 2, 0, 2, 0, 2, 0, 2, 1}
4
Returns: 10
 
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.