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 AIME Problems, 2

Tags: AMC , AIME , number theory , modular arithmetic , 2013 , AIME I

Find the number of five-digit positive integers, $n$, that satisfy the following conditions: (a) the number $n$ is divisible by $5$, (b) the first and last digits of $n$ are equal, and (c) the sum of the digits of $n$ is divisible by $5$.

2013 MTRP Senior, 8

Tags: MTRP , 2013

Suppose 51 numbers are chosen from 1, 2, 3, ..., 99, 100. Show that there are two such that one divides the other.

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 All-Russian Olympiad, 1

Tags: modular arithmetic , combinatorics proposed , combinatorics , Russia , 2013 , #5 , graph theory

$2n$ real numbers with a positive sum are aligned in a circle. For each of the numbers, we can see there are two sets of $n$ numbers such that this number is on the end. Prove that at least one of the numbers has a positive sum for both of these two sets.

2013 Greece National Olympiad, 2

Tags: quadratics , number theory proposed , number theory , Greece , 2013 , #2 , Diophantine equation

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

2013 MTRP Senior, 6

Tags: MTRP , 2013

Let N = {1, 2, . . . , n} be a set of elements called voters. Let C = {S : S $\subseteq$ N} be the power set of N. Members of C are called coalitions. Let f be a function from C to {0, 1}. A coalition S $\subseteq$ N is said to be winning if f(S) = 1; it is said to be losing if f(S) = 0. Such a function is called a voting game if the following conditions hold: (a) N is a wining coalition. (b) The empty set $\Phi$ is a losing coalition. (c) If S is a winning coalition and S $\subseteq$ S' is also winning. (d) If both S and S' are winning then S $\cap$ S' $\neq$ $\Phi$, i.e S and S' have a common voter. Show that the maximum number of winning coalitions of a voting game is $2^{n-1}$. Also find such a voting game.

2013 ELMO Shortlist, 7

Tags: function , induction , algebra unsolved , algebra , 2013 , 3n-3 theorem , continuity

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2013 ELMO Shortlist, 3

Tags: pigeonhole principle , probability , expected value , combinatorics , Elmo , 2013

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]

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 AIME Problems, 1

Tags: AMC , AIME , 2013 , AIME I , #1 , algebra

The AIME Triathlon consists of a half-mile swim, a $30$-mile bicycle, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?

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 ELMO Problems, 6

Tags: function , induction , algebra unsolved , algebra , 2013 , 3n-3 theorem , continuity

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2013 MTRP Senior, 4

Tags: MTRP , 2013

Let n be an integer such that if d | n then d + 1 | n + 1. Show that n is a prime number.

2013 MTRP Senior, 5

Tags: MTRP , 2013

A function f : $R$ $\rightarrow$ $R$ satisfies the property $f(x^2) - f^2(x) \geq 1/4$ for all x. Verify if the function is one-one.

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 MTRP Senior, 7

Tags: MTRP , 2013

Write 11 numbers on a sheet of paper six zeros and five ones. Perform the following operation 10 times: cross out any two numbers, and if they were equal, write another zero on the board. If they were not equal, write a one. Show that no matter which numbers are chosen at each step, the nal number on the board will be a one.

2013 USA Team Selection Test, 1

Tags: combinatorics , USA , TST , 2013 , graph theory

A social club has $2k+1$ members, each of whom is fluent in the same $k$ languages. Any pair of members always talk to each other in only one language. Suppose that there were no three members such that they use only one language among them. Let $A$ be the number of three-member subsets such that the three distinct pairs among them use different languages. Find the maximum possible value of $A$.

2013 MTRP Senior, 3

Tags: MTRP , 2013

Figure 1 shows a road-map connecting 14 cities. Is there a path passing through each city exactly once?

2013 MTRP Senior, 1

Tags: MTRP , 2013

Find how many committees with a chairman can be chosen from a set of n persons. Hence or otherwise prove that $${n \choose 1} + 2{n \choose 2} + 3{n \choose 3} + ...... + n{n \choose n} = n2^{n-1}$$

2013 ELMO Shortlist, 7

Tags: function , induction , algebra unsolved , algebra , 2013 , 3n-3 theorem , continuity

Consider a function $f: \mathbb Z \to \mathbb Z$ such that for every integer $n \ge 0$, there are at most $0.001n^2$ pairs of integers $(x,y)$ for which $f(x+y) \neq f(x)+f(y)$ and $\max\{ \lvert x \rvert, \lvert y \rvert \} \le n$. Is it possible that for some integer $n \ge 0$, there are more than $n$ integers $a$ such that $f(a) \neq a \cdot f(1)$ and $\lvert a \rvert \le n$? [i]Proposed by David Yang[/i]

2013 ELMO Shortlist, 3

Tags: pigeonhole principle , probability , expected value , combinatorics , Elmo , 2013

Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$? [i]Proposed by Ray Li[/i]