Found problems: 583
2007 India Regional Mathematical Olympiad, 2
Let $ a, b, c$ be three natural numbers such that $ a < b < c$ and $ gcd (c \minus{} a, c \minus{} b) \equal{} 1$. Suppose there exists an integer $ d$ such that $ a \plus{} d, b \plus{} d, c \plus{} d$ form the sides of a right-angled triangle. Prove that there exist integers, $ l,m$ such that $ c \plus{} d \equal{} l^{2} \plus{} m^{2} .$
[b][Weightage 17/100][/b]
2016 Ukraine Team Selection Test, 12
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 $n \ge N$.
1990 Vietnam Team Selection Test, 1
Let $ T$ be a finite set of positive integers, satisfying the following conditions:
1. For any two elements of $ T$, their greatest common divisor and their least common multiple are also elements of $ T$.
2. For any element $ x$ of $ T$, there exists an element $ x'$ of $ T$ such that $ x$ and $ x'$ are relatively prime, and their least common multiple is the largest number in $ T$.
For each such set $ T$, denote by $ s(T)$ its number of elements. It is known that $ s(T) < 1990$; find the largest value $ s(T)$ may take.
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2013 Online Math Open Problems, 39
Find the number of 8-digit base-6 positive integers $(a_1a_2a_3a_4a_5a_6a_7a_8)_6$ (with leading zeros permitted) such that $(a_1a_2\ldots a_8)_6\mid(a_{i+1}a_{i+2}\ldots a_{i+8})_6$ for $i=1,2,\ldots,7$, where indices are taken modulo $8$ (so $a_9=a_1$, $a_{10}=a_2$, and so on).
[i]Victor Wang[/i]
2013 IFYM, Sozopol, 4
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.
2006 Romania National Olympiad, 4
Let $A$ be a set of positive integers with at least 2 elements. It is given that for any numbers $a>b$, $a,b \in A$ we have $\frac{ [a,b] }{ a- b } \in A$, where by $[a,b]$ we have denoted the least common multiple of $a$ and $b$. Prove that the set $A$ has [i]exactly[/i] two elements.
[i]Marius Gherghu, Slatina[/i]
1995 Brazil National Olympiad, 3
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which
\[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]
2024 Macedonian TST, Problem 6
Let \(a,b\) be positive integers such that \(a+1\), \(b+1\), and \(ab\) are perfect squares. Prove that $\gcd(a,b)+1$ is also a perfect square.
2004 IMO Shortlist, 2
The function $f$ from the set $\mathbb{N}$ of positive integers into itself is defined by the equality \[f(n)=\sum_{k=1}^{n} \gcd(k,n),\qquad n\in \mathbb{N}.\]
a) Prove that $f(mn)=f(m)f(n)$ for every two relatively prime ${m,n\in\mathbb{N}}$.
b) Prove that for each $a\in\mathbb{N}$ the equation $f(x)=ax$ has a solution.
c) Find all ${a\in\mathbb{N}}$ such that the equation $f(x)=ax$ has a unique solution.
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 $x,y$.
1987 IMO Shortlist, 8
(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]
2007 AIME Problems, 8
The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$?
1987 IMO Longlists, 13
Let $A$ be an infinite set of positive integers such that every $n \in A$ is the product of at most $1987$ prime numbers. Prove that there is an infinite set $B \subset A$ and a number $p$ such that the greatest common divisor of any two distinct numbers in $B$ is $p.$
2007 All-Russian Olympiad Regional Round, 10.7
Given an integer $ n>6$. Consider those integers $ k\in (n(n\minus{}1),n^{2})$ which are coprime with $ n$. Prove that the greatest common divisor of the considered numbers is $ 1$.
2002 India IMO Training Camp, 9
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
2000 France Team Selection Test, 3
Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.
2019 Dutch IMO TST, 3
Let $n$ be a positive integer. Determine the maximum value of $gcd(a, b) + gcd(b, c) + gcd(c, a)$ for positive integers $a, b, c$ such that $a + b + c = 5n$.
2004 South africa National Olympiad, 1
Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.
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".
1988 Canada National Olympiad, 1
For what real values of $k$ do $1988x^2 + kx + 8891$ and $8891x^2 + kx + 1988$ have a common zero?
2023 Bulgaria JBMO TST, 2
Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.
2012 IMAC Arhimede, 1
Let $a_1,a_2,..., a_n$ be different integers and let $(b_1,b_2,..., b_n),(c_1,c_2,..., c_n)$ be two of their permutations, different from the identity. Prove that
$$(|a_1-b_1|+|a_2-b_2|+...+|a_n-b_n| , |a_1-c_1|+|a_2-c_2|+...+|a_n-c_n| ) \ge 2$$
where $(x,y)$ denotes the greatest common divisor of the numbers $x,y$
1966 Dutch Mathematical Olympiad, 2
For all $n$, $t_{n+1} = 2(t_n)^2 - 1$. Prove that gcd $(t_n,t_m) = 1$ if $n \ne m$.
2002 Turkey Junior National Olympiad, 3
Find all ordered positive integer pairs of $(m,n)$ such that $2^n-1$ divides $2^m+1$.