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Combinatorics
1.
(18 p.)
There are 27 candidates in elections and \( n \) citizens that vote for them. If a candidate gets \( m \) votes, then \( 100m/n \leq m1 \). What is the smallest possible value of \( n \)?
2.
(3 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).
3.
(22 p.)
At the basement of a building with 5 floors, Adam, Bob, Cindy, Diana and Ernest entered the elevator. The elevator goes only up and doesn’t come back, and each person gets out of the elevator at one of the five floors. In how many ways can the five people leave the elevator in such a way that at no time are there a male and a female alone in the elevator?
4.
(18 p.)
We are given an unfair coin. When the coin is tossed, the probability of heads is 0.4. The coin is tossed 10 times. Let \( a_n \) be the number of heads in the first \( n \) tosses. Let \( P \) be the probability that \( a_n/n \leq 0.4 \) for \( n = 1, 2, \dots , 9 \) and \( a_{10}/10 = 0.4 \). Evaluate \( \frac{P\cdot 10^{10}}{24^4} \).
5.
(37 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 \( m+n \).
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