Found problems: 583
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
Given a prime $p$, consider integers $0<a<b<c<d<p$ such that $a^4\equiv b^4\equiv c^4\equiv d^4\pmod{p}$. Show that \[a+b+c+d\mid a^{2013}+b^{2013}+c^{2013}+d^{2013}\]
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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
Let $f(x) = x-\tfrac1{x}$, and defi ne $f^1(x) = f(x)$ and $f^n(x) = f(f^{n-1}(x))$ for $n\ge2$. For each $n$, there is a minimal degree $d_n$ such that there exist polynomials $p$ and $q$ with $f^n(x) = \tfrac{p(x)}{q(x)}$ and the degree of $q$ is equal to $d_n$. Find $d_n$.
Prove that for all integers $ m$ and $ n$, the inequality
\[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\]
holds.
[i]Nanang Susyanto, Jogjakarta [/i]
Let $p \ge 2$ be a prime number and $\frac{a_p}{b_p}= 1 +\frac12+ .. +\frac{1}{p^2 -1}$, where $a_p$ and $b_p$ are two relatively prime positive integers. Compute gcd $(p, b_p)$.
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
Let $p,n$ be positive integers, such that $p$ is prime and $p <n$.
If $p$ divides $n + 1$ and $ \left(\left[\frac{n}{p}\right], (p-1)!\right) = 1$, then prove that $p\cdot \left[\frac{n}{p}\right]^2$ divides ${n \choose p} -\left[\frac{n}{p}\right]$ .
(Here $[x]$ represents the integer part of the real number $x$.)
Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$.
[i]Proposed by Morteza Saghafian, Iran[/i]
Suppose that $ k,l $ are natural numbers such that $ \gcd (11m-1,k)=\gcd (11m-1, l) , $ for any natural number $ m. $
Prove that there exists an integer $ n $ such that $ k=11^nl. $
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$
for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.
[i]Ankan Bhattacharya[/i]
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?
Prove that the equation $3y^2 = x^4 + x$ has no positive integer solutions.
Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$.
Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$.
[i]Proposed by Aaron Lin
[/i]
Find all nonnegative integers $m$ such that
\[a_m=(2^{2m+1})^2+1 \]
is divisible by at most two different primes.
We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices.
a) Prove that $\frac23E\le V$.
b) Can $E=V$?
Suppose that $S=\{a_{1}, \cdots, a_{r}\}$ is a set of positive integers, and let $S_{k}$ denote the set of subsets of $S$ with $k$ elements. Show that \[\text{lcm}(a_{1}, \cdots, a_{r})=\prod_{i=1}^{r}\prod_{s\in S_{i}}\gcd(s)^{\left((-1)^{i}\right)}.\]
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?
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
Suppose that necklace $\, A \,$ has 14 beads and necklace $\, B \,$ has 19. Prove that for any odd integer $n \geq 1$, there is a way to number each of the 33 beads with an integer from the sequence \[ \{ n, n+1, n+2, \dots, n+32 \} \] so that each integer is used once, and adjacent beads correspond to relatively prime integers. (Here a ``necklace'' is viewed as a circle in which each bead is adjacent to two other beads.)
Let $x,y$ and $a_0, a_1, a_2, \cdots $ be integers satisfying $a_0 = a_1 = 0$, and $$a_{n+2} = xa_{n+1}+ya_n+1$$for all integers $n \geq 0$. Let $p$ be any prime number. Show that $\gcd(a_p,a_{p+1})$ is either equal to $1$ or greater than $\sqrt{p}$.
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions:
$\bullet$ gcd $(a, b, c)=1$,
$\bullet$ gcd $(a, b + c)>1$,
$\bullet$ gcd $(b, c + a)>1$,
$\bullet$ gcd $(c, a + b)>1$.
a) Is it possible that $a + b + c = 2015$?
b) Determine the minimum possible value that the sum $a+ b+ c$ can take.