Olympiad problems

2013 Stanford Mathematics Tournament, 2

Consider the numbers $\{24,27,55,64,x\}$. Given that the mean of these five numbers is prime and the median is a multiple of $3$, compute the sum of all possible positive integral values of $x$.

2013 Stanford Mathematics Tournament, 8

Tags: 2013 , Advanced Topics

Farmer John owns 2013 cows. Some cows are enemies of each other, and Farmer John wishes to divide them into as few groups as possible such that each cow has at most 3 enemies in her group. Each cow has at most 61 enemies. Compute the smallest integer $G$ such that, no matter which enemies they have, the cows can always be divided into at most $G$ such groups?

2013 Stanford Mathematics Tournament, 4

Tags: 2013 , Advanced Topics

Given the digits $1$ through $7$, one can form $7!=5040$ numbers by forming different permutations of the $7$ digits (for example, $1234567$ and $6321475$ are two such permutations). If the $5040$ numbers are then placed in ascending order, what is the $2013^{\text{th}}$ number?

2013 Stanford Mathematics Tournament, 5

Tags: 2013 , Advanced Topics , probability , expected value

An unfair coin lands heads with probability $\tfrac1{17}$ and tails with probability $\tfrac{16}{17}$. Matt flips the coin repeatedly until he flips at least one head and at least one tail. What is the expected number of times that Matt flips the coin?

2013 Stanford Mathematics Tournament, 6

Tags: 2013 , Advanced Topics

A positive integer $b\geq 2$ is [i]neat[/i] if and only if there exist positive base-$b$ digits $x$ and $y$ (that is, $x$ and $y$ are integers, $0<x<b$ and $0<y<b$) such that the number $x.y$ base $b$ (that is, $x+\tfrac yb$) is an integer multiple of $x/y$. Find the number of [i]neat[/i] integers less than or equal to $100$.

2013 Stanford Mathematics Tournament, 9

Tags: 2013 , Advanced Topics

Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.

2013 Stanford Mathematics Tournament, 7

Tags: 2013 , Advanced Topics

Robin is playing notes on an 88-key piano. He starts by playing middle C, which is actually the 40th lowest note on the piano (i.e. there are 39 notes lower than middle C). After playing a note, Robin plays with probability $\tfrac12$ the lowest note that is higher than the note he just played, and with probability $\tfrac12$ the highest note that is lower than the note he just played. What is the probability that he plays the highest note on the piano before playing the lowest note?

2013 Stanford Mathematics Tournament, 3

Tags: 2013 , Advanced Topics

Nick has a terrible sleep schedule. He randomly picks a time between 4 AM and 6 AM to fall asleep, and wakes up at a random time between 11 AM and 1 PM of the same day. What is the probability that Nick gets between 6 and 7 hours of sleep?

2013 Stanford Mathematics Tournament, 1

Tags: 2013 , Advanced Topics

How many positive three-digit integers $\underline a\, \underline b\,\underline c$ can represent a valid date in $2013$, where either $a$ corresponds to a month and $\underline b\,\underline c$ corresponds to the day in that month, or $\underline a\, \underline b$ corresponds to a month and $c$ corresponds to the day? For example, 202 is a valid representation for February 2nd, and 121 could represent either January 21st or December 1st.