The numbers in the sequence 101, 104, 109, 116, $\dots$ are of the form $a_n = 100 + n^2$, where $n = 1$, 2, 3, $\dots$. For each $n$, let $d_n$ be the greatest common divisor of $a_n$ and $a_{n + 1}$. Find the maximum value of $d_n$ as $n$ ranges through the positive integers.

For a prime $q$, let $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$. Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying \[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \][i]Proposed by Sammy Luo[/i]

Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.

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]

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Suppose $a$, $b$, and $c$ are positive integers such that \[\frac{a}{14}+\frac{b}{15}=\frac{c}{210}.\] Which of the following statements are necessarily true? I. If $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both, then $\gcd(c,210)=1$. II. If $\gcd(c,210)=1$, then $\gcd(a,14)=1$ or $\gcd(b,15)=1$ or both. III. $\gcd(c,210)=1$ if and only if $\gcd(a,14)=\gcd(b,15)=1$. $\textbf{(A)}~\text{I, II, and III}\qquad\textbf{(B)}~\text{I only}\qquad\textbf{(C)}~\text{I and II only}\qquad\textbf{(D)}~\text{III only}\qquad\textbf{(E)}~\text{II and III only}$

Let $a_i, b_i$ be coprime positive integers for $i = 1, 2, \ldots , k$, and $m$ the least common multiple of $b_1, \ldots , b_k$. Prove that the greatest common divisor of $a_1 \frac{m}{b_1} , \ldots, a_k \frac{m}{b_k}$ equals the greatest common divisor of $a_1, \ldots , a_k.$

For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.

Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.

The Fibonacci sequence $\{F_{n}\}$ is defined by \[F_{1}=1, \; F_{2}=1, \; F_{n+2}=F_{n+1}+F_{n}.\] Show that $\gcd (F_{m}, F_{n})=F_{\gcd (m, n)}$ for all $m, n \in \mathbb{N}$.

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.

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + mx -1 = 0$ where $m$ is an odd integer. Let $\lambda _n = \alpha ^n + \beta ^n , n \geq 0$ Prove that (A) $\lambda _n$ is an integer (B) gcd ( $\lambda _n , \lambda_{n+1}$) = 1 .

Determine all integer $n > 1$ such that \[\gcd \left( n, \dfrac{n-m}{\gcd(n,m)} \right) = 1\] for all integer $1 \le m < n$.

For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for any pair of naturals $m,n$, $$\gcd(f(m),n) = \gcd(m,f(n)).$$

Suppose that both $x^{3}-x$ and $x^{4}-x$ are integers for some real number $x$. Show that $x$ is an integer.

Determine all triples of positive integers $a, b, c$ that satisfy a) $[a, b] + [a, c] + [b, c] = [a, b, c]$. b) $[a, b] + [a, c] + [b, c] = [a, b, c] + (a, b, c)$. Remark: Here $[x, y$] denotes the least common multiple of positive integers $x$ and $y$, and $(x, y)$ denotes their greatest common divisor.

In how many ways can you divide the set of eight numbers $\{2,3,\cdots,9\}$ into $4$ pairs such that no pair of numbers has $\text{gcd}$ equal to $2$?

Suppose that $a_0, a_1, \cdots $ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and \[ a_{n+1} = \gcd{(a_n, b_n)} + 1, \qquad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1. \] Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.

Integers $a, b, c, d$ satisfy the following: $abcd=2^6\cdot 3^9\cdot 5^7$ $\text{lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$ $\text{lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$ $\text{lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$ $\text{lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$ $\text{lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$ $\text{lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$ Find $\text{gcd}(a,b,c,d)$ $\textbf{(A)}~30\qquad\textbf{(B)}~45\qquad\textbf{(C)}~3\qquad\textbf{(D)}~15\qquad\textbf{(E)}~6$

Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained? \\

Let $a, b,$ and $c$ be positive integers such that $a+b+c=23$ and \[\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.\] What is the sum of all possible distinct values of $a^{2}+b^{2}+c^{2}$? $\textbf{(A)} ~259\qquad\textbf{(B)} ~438\qquad\textbf{(C)} ~516\qquad\textbf{(D)} ~625\qquad\textbf{(E)} ~687$ Proposed by [b]djmathman[/b]

The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get?

Solve the equation $mn =$ (gcd($m,n$))$^2$ + lcm($m, n$) in positive integers, where gcd($m, n$) – greatest common divisor of $m,n$, and lcm($m, n$) – least common multiple of $m,n$.