IMOmath

Combinatorics

1. (8 p.)
Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.

2. (22 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 \( (5,5) \)?

3. (26 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. (19 p.)
A circle of radius 1 is randomly placed inside a \( 15 \times 36 \) rectangle \( ABCD \). The probability that it does not intersect the diagonal \( AC \) can be expressed as \( p/q \) where \( p \) and \( q \) are relatively prime integers. Find \( p+q \).

5. (22 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 \( p+q \).





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