Found problems: 583
2020 AMC 12/AHSME, 21
How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$
$\textbf{(A) } 12 \qquad \textbf{(B) } 24 \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 72$
1987 India National Olympiad, 6
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
2014 Contests, 1
Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.
2012 Albania Team Selection Test, 4
Find all couples of natural numbers $(a,b)$ not relatively prime ($\gcd(a,b)\neq\ 1$) such that
\[\gcd(a,b)+9\operatorname{lcm}[a,b]+9(a+b)=7ab.\]
2004 Iran MO (3rd Round), 21
$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$
\[ p\mid a_1^k \plus{} \dots \plus{} a_n^k.\]
1990 IMO Longlists, 57
The sequence $\{u_n\}$ is defined by $u_1 = 1, u_2 = 1, u_n = u_{n-1} + 2u_{n-2} for n \geq 3$. Prove that for any positive integers $n, p \ (p > 1), u_{n+p} = u_{n+1}u_{p} + 2u_nu_{p-1}$. Also find the greatest common divisor of $u_n$ and $u_{n+3}.$
2005 Brazil National Olympiad, 6
Given positive integers $a,c$ and integer $b$, prove that there exists a positive integer $x$ such that
\[ a^x + x \equiv b \pmod c, \]
that is, there exists a positive integer $x$ such that $c$ is a divisor of $a^x + x - b$.
2006 China Team Selection Test, 3
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
2010 Princeton University Math Competition, 1
Show that the GCD of three consecutive triangular numbers is 1.
2013 Peru MO (ONEM), 2
The positive integers $a, b, c$ are such that
$$gcd \,\,\, (a, b, c) = 1,$$
$$gcd \,\,\,(a, b + c) > 1,$$
$$gcd \,\,\,(b, c + a) > 1,$$
$$gcd \,\,\,(c, a + b) > 1.$$
Determine the smallest possible value of $a + b + c$.
Clarification: gcd stands for greatest common divisor.
2022 JHMT HS, 10
Compute the exact value of
\[ \sum_{a=1}^{\infty}\sum_{b=1}^{\infty} \frac{\gcd(a, b)}{(a + b)^3}. \]
If necessary, you may express your answer in terms of the Riemann zeta function, $Z(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$, for integers $s \geq 2$.
2002 Bosnia Herzegovina Team Selection Test, 3
If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.
2016 India IMO Training Camp, 3
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.
1995 India National Olympiad, 6
Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.
2007 China National Olympiad, 2
Show that:
1) If $2n-1$ is a prime number, then for any $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, there exists $i, j \in \{1, 2, \ldots , n\}$ such that
\[\frac{a_i+a_j}{(a_i,a_j)} \geq 2n-1\]
2) If $2n-1$ is a composite number, then there exists $n$ pairwise distinct positive integers $a_1, a_2, \ldots , a_n$, such that for any $i, j \in \{1, 2, \ldots , n\}$ we have
\[\frac{a_i+a_j}{(a_i,a_j)} < 2n-1\]
Here $(x,y)$ denotes the greatest common divisor of $x,y$.
2022 Mexican Girls' Contest, 1
Determine all finite nonempty sets $S$ of positive integers satisfying
\[ {i+j\over (i,j)}\qquad\mbox{is an element of S for all i,j in S}, \]
where $(i,j)$ is the greatest common divisor of $i$ and $j$.
2016 Switzerland Team Selection Test, Problem 5
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.
1980 IMO, 1
Let $p(x)$ be a polynomial with integer coefficients such that $p(0)=p(1)=1$. We define the sequence $a_0, a_1, a_2, \ldots, a_n, \ldots$ that starts with an arbitrary nonzero integer $a_0$ and satisfies $a_{n+1}=p(a_n)$ for all $n \in \mathbb N\cup \{0\}$. Prove that $\gcd(a_i,a_j)=1$ for all $i,j \in \mathbb N \cup \{0\}$.
1998 Hong kong National Olympiad, 3
Given $s,t$ are non-zero integers, $(x,y) $ is an integer pair , A transformation is to change pair $(x,y)$ into pair $(x+t,y-s)$ . If the two integers in a certain pair becoems relatively prime after several tranfomations , then we call the original integer pair "a good pair" .
(1) Is $(s,t)$ a good pair ?
(2) Prove :for any $s$ and $t$ , there exists pair $(x,y)$ which is " a good pair".
1997 Pre-Preparation Course Examination, 1
Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that
\[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]
1997 Poland - Second Round, 4
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.
1979 IMO Longlists, 55
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.
2022 Thailand TSTST, 2
Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
1988 Mexico National Olympiad, 5
If $a$ and $b$ are coprime positive integers and $n$ an integer, prove that the greatest common divisor of $a^2+b^2-nab$ and $a+b$ divides $n+2$.
2004 USAMO, 2
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.