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Combinatorics
1.
(10 p.)
Assume that \( A \) is a 40element subset of \( \{1,2,3,\dots,50\} \), and let \( n \) be the sum of the elements of \( A \). Find the number of possible values of \( n \).
2.
(14 p.)
Given a convex polyhedron with 26 vertices, 60 edges and 36 faces, 24 of the faces are triangular and 12 are quadrilaterals. A space diagonal is a line segment connecting two vertices which do not belong to the same face. How many space diagonals does the polyhedron have?
3.
(50 p.)
Bob is making partitions of \( 10 \), but he hates even numbers, so he splits \( 10 \) up in a special way. He starts with \( 10 \), and at each step he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, \( 6 \) could be replaced with \( 1+5 \), \( 2+4 \), or \( 3+3 \) all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).
4.
(22 p.)
In a tournament club \( C \) plays 6 matches, and for each match the probabilities of a win, draw and loss are equal. If the probability that \( C \) finishes with more wins than losses is \( \frac pq \) with \( p \) and \( q \) coprime \( (q>0) \), find \( p+q \).
5.
(4 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).
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