IMOmath

Combinatorics

1. (16 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?

2. (35 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?

3. (30 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 \( p+q \).

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

5. (11 p.)
Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.





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