Found problems: 583
2018 Romania Team Selection Tests, 4
Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.
1988 ITAMO, 7
Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.
2009 Putnam, A4
Let $ S$ be a set of rational numbers such that
(a) $ 0\in S;$
(b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and
(c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$
Must $ S$ contain all rational numbers?
1953 Moscow Mathematical Olympiad, 244
Prove that $gcd (a + b, lcm(a, b)) = gcd (a, b)$ for any $a, b$.
2015 Mathematical Talent Reward Programme, SAQ: P 5
Let $a$ be the smallest and $A$ the largest of $n$ distinct positive integers. Prove that the least common multiple of these numbers is greater than or equal to $n a$ and that the greatest common divisor is less than or equal to $\frac{A}{n}$
2019 Nigeria Senior MO Round 2, 4
Let $h(t)$ and $f(t)$ be polynomials such that $h(t)=t^2$ and $f_n(t)=h(h(h(h(h...h(t))))))-1$ where $h(t)$ occurs $n$ times. Prove that $f_n(t)$ is a factor of $f_N(t)$ whenever $n$ is a factor of $N$
2006 IMO Shortlist, 6
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]
1998 Federal Competition For Advanced Students, Part 2, 2
Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$.
[b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$.
[b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.
2014 Contests, 3
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
2014 Baltic Way, 19
Let $m$ and $n$ be relatively prime positive integers. Determine all possible values of \[\gcd(2^m - 2^n, 2^{m^2+mn+n^2}- 1).\]
2017 Pan-African Shortlist, N1
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
1991 Romania Team Selection Test, 8
Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$.
If $a = 1$, show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$.
1970 AMC 12/AHSME, 34
The greatest integer that will divide $13,511$, $13,903$, and $14,589$ and leave the same remainder is
$\textbf{(A) }28\qquad\textbf{(B) }49\qquad\textbf{(C) }98\qquad$
$\textbf{(D) }\text{an odd multiple of }7\text{ greater than }49\qquad \textbf{(E) }\text{an even multiple of }7\text{ greater than }98$
2022 Germany Team Selection Test, 1
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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2005 Taiwan TST Round 1, 3
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!)
2011 Korea National Olympiad, 2
Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.
2022 Bolivia IMO TST, P4
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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
2014 Argentina National Olympiad Level 2, 6
Let $a, b, c$ be distinct positive integers with sum $547$ and let $d$ be the greatest common divisor of the three numbers $ab+1, bc+1, ca+1$. Find the maximal possible value of $d$.
2023 Bulgaria EGMO TST, 2
Determine all integers $k$ for which there exists a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}$ such that $f(2023) = 2024$ and $f(ab) = f(a) + f(b) + kf(\gcd(a,b))$ for all positive integers $a$ and $b$.
2002 Tournament Of Towns, 1
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
2013 IFYM, Sozopol, 4
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
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.
2024 Ukraine National Mathematical Olympiad, Problem 1
Find all pairs $a, b$ of positive integers, for which
$$(a, b) + 3[a, b] = a^3 - b^3$$
Here $(a, b)$ denotes the greatest common divisor of $a, b$, and $[a, b]$ denotes the least common multiple of $a, b$.
[i]Proposed by Oleksiy Masalitin[/i]
2020 Federal Competition For Advanced Students, P1, 3
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)
2018-IMOC, N2
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$\operatorname{lcm}(f(x),y)\gcd(f(x),f(y))=f(x)f(f(y))$$
for all $x,y\in\mathbb N$.