Determine if there is a set $S$ of 2011 positive integers so that for every pair $m,n$ of distinct elements of $S$, $|m-n|=(m,n)$. Here $(m,n)$ denotes the greatest common divisor of $m$ and $n$.

Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in every $k$-subset of $S$, having the above mentioned properties there are two distinct elements with greatest common divisor greater than 1.

For positive integers $m$ and $n$ we define $T(m,n) = gcd \left(m, \frac{n}{gcd(m,n)} \right)$ (a) Prove that there are infinitely many pairs $(m,n)$ of positive integers for which $T(m,n) > 1$ and $T(n,m) > 1$. (b) Do there exist positive integers $m,n$ such that $T(m,n) = T(n,m) > 1$?

Given positive integer $n$ and $r$ pairwise distinct primes $p_1,p_2,\cdots,p_r.$ Initially, there are $(n+1)^r$ numbers written on the blackboard: $p_1^{i_1}p_2^{i_2}\cdots p_r^{i_r} (0 \le i_1,i_2,\cdots,i_r \le n).$ Alice and Bob play a game by making a move by turns, with Alice going first. In Alice's round, she erases two numbers $a,b$ (not necessarily different) and write $\gcd(a,b)$. In Bob's round, he erases two numbers $a,b$ (not necessarily different) and write $\mathrm{lcm} (a,b)$. The game ends when only one number remains on the blackboard. Determine the minimal possible $M$ such that Alice could guarantee the remaining number no greater than $M$, regardless of Bob's move.

We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$. a) Are there infinitely many lovely numbers? b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?

Set $S_n = \sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$

Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.

Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]

$a,b$ are naturals and $$a \times GCD(a,b)+b \times LCM(a,b)<2.5 ab$$. Prove that $b|a$

Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$. [i]Mihai Pitticari & Sorin Rǎdulescu[/i]

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

On a blackboard there are three positive integers. In each step the three numbers on the board are denoted as $a, b, c$ such that $a >gcd(b, c)$, then $a$ gets replaced by $ a-gcd(b, c)$. The game ends if there is no way to denote the numbers such that $a >gcd(b, c)$. Prove that the game always ends and that the last three numbers on the blackboard only depend on the starting numbers. (Theresia Eisenkölbl)

For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?

Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| \plus{} |y|$ taken over all pairs of integers $ x$ and $ y$ such that \[ax \plus{} by \equal{} c.\] An integer $ c$ is called a [i]local champion [/i]if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$. Find all local champions and determine their number. [i]Proposed by Zoran Sunic, USA[/i]

(a) Let $\gcd(m, k) = 1$. Prove that there exist integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ such that each product $a_ib_j$ ($i = 1, 2, \cdots ,m; \ j = 1, 2, \cdots, k$) gives a different residue when divided by $mk.$ (b) Let $\gcd(m, k) > 1$. Prove that for any integers $a_1, a_2, . . . , a_m$ and $b_1, b_2, . . . , b_k$ there must be two products $a_ib_j$ and $a_sb_t$ ($(i, j) \neq (s, t)$) that give the same residue when divided by $mk.$ [i]Proposed by Hungary.[/i]

let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Let $S = \{1, 2, 3, 5, 6, 10, 15, 30\}$. For each of the $64$ ordered pairs $(a, b)$ of elements of $S$, AJ computes $gcd(a, b)$. They then sum all of the numbers they computed. What is AJ’s sum?

Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.

Find all positive integer triples $(x,y,z)$ such that $x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$. Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)

Find all triples $(a, b, c)$ of positive integers for which $a + (a, b) = b + (b, c) = c + (c, a)$. Here $(a, b)$ denotes the greatest common divisor of integers $a, b$. [i](Proposed by Mykhailo Shtandenko)[/i]

Find all sets $S$ of positive integers that satisfy all of the following. $1.$ If $a,b$ are two not necessarily distinct elements in $S$, then $\gcd(a,b)$, $ab$ are also in $S$. $2.$ If $m,n$ are two positive integers with $n\nmid m$, then there exists an element $s$ in $S$ such that $m^2\mid s$ and $n^2\nmid s$. $3.$ For any odd prime $p$, the set formed by moduloing all elements in $S$ by $p$ has size exactly $\frac{p+1}2$.

Suppose $\, q_{0}, \, q_{1}, \, q_{2}, \ldots \; \,$ is an infinite sequence of integers satisfying the following two conditions: (i) $\, m-n \,$ divides $\, q_{m}-q_{n}\,$ for $\, m > n \geq 0,$ (ii) there is a polynomial $\, P \,$ such that $\, |q_{n}| < P(n) \,$ for all $\, n$ Prove that there is a polynomial $\, Q \,$ such that $\, q_{n}= Q(n) \,$ for all $\, n$.

Let $P$ be a polynomial with integer coefficients such that $P(0)=0$ and \[\gcd(P(0), P(1), P(2), \ldots ) = 1.\] Show there are infinitely many $n$ such that \[\gcd(P(n)- P(0), P(n+1)-P(1), P(n+2)-P(2), \ldots) = n.\]