Log In
Register
IMOmath
Olympiads
Book
Training
IMO Results
Forum
IMOmath
Combinatorics
1.
(10 p.)
Two students Alice and Bob participated in a twoday 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 twoday success ratio was 300/500 = 3/5. Bob’s scores are different from Alice’s (but with the same twoday 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 twoday success ratio that Bob could have achieved. Calculate \( p+q \).
2.
(10 p.)
Given a regular 12gon D, determine the number of squares that have two or more vertices among the vertices of D.
3.
(26 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.
(24 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 \( \frac{P\cdot 10^{10}}{24^4} \).
5.
(30 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?
20052018
IMOmath.com
 imomath"at"gmail.com  Math rendered by
MathJax
Home

Olympiads

Book

Training

IMO Results

Forum

Links

About

Contact us