IMOmath

Combinatorics

1. (8 p.)
Two students Alice and Bob participated in a two-day math contest. At the end both had attempted questions worth 500 points. Alice scored 160 out of 300 attempted on the first day and 140 out of 200 attempted on the second day, so her two-day success ratio was 300/500 = 3/5. Bob’s scores are different from Alice’s (but with the same two-day total). Bob had a positive integer score on each day. However, for each day Bob’s success ratio was less than Alice’s. Assume that \( p/q \) (\( p \) and \( q \) are relatively prime integers) is the largest possible two-day success ratio that Bob could have achieved. Calculate \( p+q \).

2. (21 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. (40 p.)
Bob is making partitions of \( 10 \), but he hates even numbers, so he splits \( 10 \) up in a special way. He starts with \( 10 \), and at each step he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, \( 6 \) could be replaced with \( 1+5 \), \( 2+4 \), or \( 3+3 \) all with equal probability. He terminates this process when all the numbers in his list are odd. The expected number of integers in his list at the end can be expressed in the form \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

4. (8 p.)
Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.

5. (21 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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