Six children stand in a line outside their classroom. When they enter the classroom, they sit in a circle in random order. There are relatively prime positive integers $m$ and $n$ so that $\tfrac{m}{n}$ is the probability that no two children who stood next to each other in the line end up sitting next to each other in the circle. Find $m + n$.

Two points $A$ and $B$ are selected independently and uniformly at random along the perimeter of a unit square with vertices at $(0,0)$, $(1,0)$, $(0,1)$, and $(1,1)$. The probability that the $y$-coordinate of $A$ is strictly greater than the $y$-coordinate of $B$ can be expressed as $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$. [i]Proposed by Rajiv Movva[/i]

Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $ ($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$. [i]Author: Alex Zhu[/i]

Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$. [i]A. Kovaldji, V. Senderov[/i]

Prove that there exists a set $X \subseteq \mathbb{N}$ which contains exactly 2022 elements such that for every distinct $a, b, c \in X$ the following equality: \[ \gcd(a^n+b^n, c) = 1 \] is satisfied for every positive integer $n$.

Determine the number of ordered pairs of positive integers $(x,y)$ with $y < x \le 100$ such that $x^2-y^2$ and $x^3 - y^3$ are relatively prime. (Two numbers are [i]relatively prime[/i] if they have no common factor other than $1$.) [i]Ray Li[/i]

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]

a) Show that the set $ \mathbb{Q}^{ + }$ of all positive rationals can be partitioned into three disjoint subsets. $ A,B,C$ satisfying the following conditions: \[ BA = B; \& B^2 = C; \& BC = A; \] where $ HK$ stands for the set $ \{hk: h \in H, k \in K\}$ for any two subsets $ H, K$ of $ \mathbb{Q}^{ + }$ and $ H^2$ stands for $ HH.$ b) Show that all positive rational cubes are in $ A$ for such a partition of $ \mathbb{Q}^{ + }.$ c) Find such a partition $ \mathbb{Q}^{ + } = A \cup B \cup C$ with the property that for no positive integer $ n \leq 34,$ both $ n$ and $ n + 1$ are in $ A,$ that is, \[ \text{min} \{n \in \mathbb{N}: n \in A, n + 1 \in A \} > 34. \]

Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.

Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is $\tfrac{1}{3}$. The probability that a man has none of the three risk factors given that he does not have risk factor A is $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Let $G$ be a hyper elementary abelian $p-$group and let $f : G \rightarrow G$ be a homomorphism. Then prove that $\ker f$ is isomorphic to $\mathrm{coker} f$.

We define the sequence $x_n$ so that \[x_1=a, x_2=b, x_n=\frac{{x_{n-1}}^2+{x_{n-2}}^2}{x_{n-1}+x_{n-2}} \quad \forall n \geq 3.\] Where $a,b >1$ are relatively prime numbers. Show that $x_n$ is not an integer for $n \geq 3$.

The perimeter of triangle $APM$ is $152,$ and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}.$ Given that $OP=m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$

Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$. [i]Proposed by Iurie Boreico[/i]

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.

A sequence of integers is defined as follows: $a_1=1,a_2=2,a_3=3$ and for $n>3$, $$a_n=\textsf{The smallest integer not occurring earlier, which is relatively prime to }a_{n-1}\textsf{ but not relatively prime to }a_{n-2}.$$Prove that every natural number occurs exactly once in this sequence. [i]M. Ivanov[/i]

Suppose $a_1, \dots, a_n$ are integers whose greatest common divisor is 1. Let $S$ be a set of integers with the following properties: (a) For $i=1, \dots, n$, $a_i \in S$. (b) For $i,j = 1, \dots, n$ (not necessarily distinct), $a_i - a_j \in S$. (c) For any integers $x,y \in S$, if $x+y \in S$, then $x-y \in S$. Prove that $S$ must be equal to the set of all integers.

Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.) [i]Proposed by Alexander S. Golovanov, Russia[/i]

Prove that if from any $2007$ consecutive terms of an infinite arithmetic progression of integers starting with $2$, one can choose a term relatively prime to all the $2006$ other terms, then there is also a term amongst any $2008$ consecutive terms relatively prime to the rest.

The number \[ \frac 2{\log_4{2000^6}}+\frac 3{\log_5{2000^6}} \] can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$