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Suppose there is a circle with \(15\) evenly spaced points numbered from \(1\) to \(15\) in clockwise order. A star is defined as a sequence five numbers \(a_1, a_2, a_3,a_4,a_5\) with \(a_1 < a_4 < a_2 < a_5 < a_3, 1 \le a_1, a_2, a_3, a_4, a_5 \le 15\) and \(\min(|a_{i + 1} - a_i|, 15 - |a_{i + 1}-a_i|) \ge 2\) for integers \(i\) with \(1\le i \le 5.\) (Let \(a_6 = a_1\).)
Find the number of possible stars.
Let \(A_1 = (0,0), A_2=(1,0), A_3=(2,0), A_4=(3,0),A_5=(4,0)\) and \(B_1 = (0,1), B_2=(1,1), B_3=(2,1), B_4=(3,1),B_5=(4,1).\) Let \(L\) be the set of line segments \(A_iB_j\) where \(i\) and \(j\) are integers with \(1 \le i, j \le 5.\) (\(i\) and \(j\) do not need to be distinct.) Let \(S\) be the set of points in the interior of rectangle \(A_1A_5B_5B_1\) that are the intersection of two distinct line segments in \(L.\)
Of all the points in \(S,\) how many different \(y\)-coordinates are there? If two points have the same \(y\)-coordinate, then that \(y\)-coordinate should only be counted once.
Let \(B\) be the set of (distinct) unordered pairs of fair 16-sided dice with a positive integer written on each face such that the likelihood that the roll a sum of \(k\) is the same as the likelihood of rolling a sum of \(k\) in a pair of fair standard 16-sided dice with faces numbered 1-16, for all \(2 \le k \le 32\). Each element of \(B\) can be represented as an unordered pair of 16-tuples.
Let \(y\) be the maximum difference between the max value of one die with the max value of the other die in any given pair of dice in \(B\).
Find \(|B|y\).
For instance, if we had the same problem but with fair 6-sided die, then \[B = \{\{(1,2,3,4,5,6), (1,2,3,4,5,6)\}, \{(1,2,2,3,3,4), (1,3,4,5,6,8)\}\} \] where the 6-tuple \((1,2,2,3,3,4)\) represents a cube with faces labeled 1, 2, 2, 3, 3, and 4. So \(|B| = 2\) and \(y = \max(8 - 4, 6- 6) = 4\) in this case.
Let \(A_1 = (0,0), A_2=(1,0), A_3=(2,0), A_4=(3,0)\) and \(B_1 = (0,1), B_2=(1,1), B_3=(2,1), B_4=(3,1).\) Let \(L\) be the set of line segments \(A_iB_j\) where \(i\) and \(j\) are integers with \(1 \le i, j \le 4.\) (\(i\) and \(j\) do not need to be distinct.) Let \(S\) be the set of points in the interior of rectangle \(A_1A_4B_4B_1\) that are the intersection of two distinct line segments in \(L.\)
Find the number of ways to choose points \(A_i,A_j,P_1\) and \(P_2,\) where \(i\) and \(j\) are integers with \(1 \le i < j \le 4,\) \(P_1\) and \(P_2\) are points in \(S\) (the order of \(P_1\) and \(P_2\) does not matter), and \(A_iP_1P_2A_j\) forms a trapezoid with \(P_1P_2 \parallel A_iA_j.\)
Due to an error, everyone will receive free points for this problem.
An octahedron has vertices at \((\pm 1, 0,0)\), \((0, \pm 1, 0)\), and \((0,0 \pm 1)\). The notation \((\pm 1, 0, 0)\) represents both points \((1,0,0)\) and \((-1,0,0)\). It is rotated about the line \(x = y = z\) by \(120\) degrees. Let \(V\) be the volume of space that the octahedron, including its interior, traces out in this process. Then \(V\) can be written in the form \(\left(\frac{a\sqrt{b} + c}{d} \right)\pi\) where \(b\) is not divisible by the square of any prime, and \(\gcd(a,c,d) = 1\). Find \(a + b + c + d\).