IMOmath

Combinatorics

1. (28 p.)
There are 27 candidates in elections and \( n \) citizens that vote for them. If a candidate gets \( m \) votes, then \( 100m/n \leq m-1 \). What is the smallest possible value of \( n \)?

2. (26 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 \( p+q \).

3. (4 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).

4. (11 p.)
Assume that \( A \) is a 40-element 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 \( n \).

5. (28 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 \( \frac{P\cdot 10^{10}}{24^4} \).





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