Log In
Register
IMOmath
Olympiads
Book
Training
Forum
IMOmath
Combinatorics
1.
(30 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 \( \frac{P\cdot 10^{10}}{24^4} \).
2.
(7 p.)
Let \( S = \{1, 2, 3, 5, 8, 13, 21, 34\} \). Find the sum \( \sum \max(A) \) where the sum is taken over all 28 twoelement subsets \( A \) of \( S \).
3.
(30 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 \( n \)?
4.
(27 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.
(5 p.)
Let \( S \) be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to \( S \).
20052020
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us