Found problems: 583
Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{b^{2}}= b^{a}.\]
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
Solve the following equation in integers with gcd (x, y) = 1
$x^2 + y^2 = 2 z^2$
Let $a,b,c\in\mathbb{Z^{+}}$ such that
$$(a^2-1, b^2-1, c^2-1)=1$$
Prove that
$$(ab+c, bc+a, ca+b)=(a,b,c)$$
(As usual, $(x,y,z)$ means the greatest common divisor of numbers $x,y,z$)
[I]Proposed by A. Golovanov[/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.
Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.
Find all non-negative integers $ x,y,z$ such that $ 5^x \plus{} 7^y \equal{} 2^z$.
:lol:
([i]Daniel Kohen, University of Buenos Aires - Buenos Aires,Argentina[/i])
Let $n$ be an odd positive integer with $n > 1$ and let $a_1, a_2,... , a_n$ be positive integers such that gcd $(a_1, a_2,... , a_n) = 1$. Let $d$ = gcd $(a_1^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, a_2^n + a_1\cdot a_2 \cdot \cdot \cdot a_n, ... , a_n^n + a_1\cdot a_2 \cdot \cdot \cdot a_n)$. Show that the possible values of $d$ are $d = 1, d = 2$
Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.
$a+b+c \leq 3000000$ and $a\neq b \neq c \neq a$ and $a,b,c$ are naturals.
Find maximum $GCD(ab+1,ac+1,bc+1)$
Let $c$ be a nonnegative integer, and define $a_n = n^2 + c$ (for $n \geq 1)$. Define $d_n$ as the greatest common divisor of $a_n$ and $a_{n + 1}$.
(a) Suppose that $c = 0$. Show that $d_n = 1,\ \forall n \geq 1$.
(b) Suppose that $c = 1$. Show that $d_n \in \{1,5\},\ \forall n \geq 1$.
(c) Show that $d_n \leq 4c + 1,\ \forall n \geq 1$.
Is it possible to assign every integral point $(x,y)$ of the plane a positive integer $a_{x,y}$ such that for every two integers $i$ and $j$ the following equality holds $$a_{i,j}=\gcd(a_{i-1,j},a_{i+1,j})+\gcd(a_{i,j-1},a_{i,j+1})$$
[i]M. Shutro[/i]
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 $a + b + c = 5n$.
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 $s \geq 2$.
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i]
Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.
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]
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 $d$.
For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.
For every natural number $a$ and $b$, define the notation $[a,b]$ as the least common multiple of $a $ and $b$ and the notation $(a,b)$ as the greatest common divisor of $a$ and $b$. Find all $n \in \mathbb{N}$ that satisfies
\[
4 \sum_{k=1}^{n} [n,k] = 1 + \sum_{k=1}^{n} (n,k) + 2n^2 \sum_{k=1}^{n} \frac{1}{(n,k)}
\]
Determine the smallest positive integer $n\geq 2$ for which there exists a positive integer $m$ such that $mn$ divides $m^{2023} + n^{2023} + n$.
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.
For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$.
Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer.
[i]Dan Schwarz, Romania[/i]
A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares.
[i]Proposed by A. Golovanov[/i]
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 $i,j \in \mathbb N \cup \{0\}$.