# Combinatorics

 1. (4 p.) Let $$S$$ be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to $$S$$.

 2. (26 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 $$\frac{P\cdot 10^{10}}{24^4}$$.

 3. (28 p.) A frog is jumping in the coordinate plane according to the following rules: (i) From any lattice point $$(a,b)$$, the frog can jump to $$(a+1,b)$$, $$(a,b+1)$$, or $$(a+1,b+1)$$. (ii) There are no right angle turns in the frog’s path. How many different paths can the frog take from $$(0,0)$$ to $$(5,5)$$?

 4. (11 p.) Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.

 5. (28 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us