Found problems: 583
Let $k,l$ be two given integers. Prove that there exist infinite many integers $m\ge k$ such that $\gcd\left(\binom{m}{k},l\right)=1$.
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.
Let $\text{lcm} (a,b)$ denote the least common multiple of $a$ and $b$. Find the sum of all positive integers $x$ such that $x\le 100$ and $\text{lcm}(16,x) = 16x$.
[i]Ray Li.[/i]
Show that $$5^{2019} \mid \Sigma^{5^{2019}}_{k=1}3^{gcd (5^{2019},k)}$$
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".
$a)$ Let $m$ and $n$ be positive integers. If $m>1$ prove that $ n \mid \phi(m^n-1)$ where $\phi$ is Euler function
$b)$ Prove that number of elements in sequence $1,2,...,n$ $(n \in \mathbb{N})$, which greatest common divisor with $n$ is $d$, is $\phi\left(\frac{n}{d}\right)$
A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x \plus{} 1$ and $ \frac {x}{x \plus{} 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.
Determine all ordered triples of positive integers $(x, y, z)$ such that $\gcd(x+y, y+z, z+x) > \gcd(x, y, z)$.
We call a positive integer $n$ [i]amazing[/i] if there exist positive integers $a, b, c$ such that the equality
\[n = (b, c)(a, bc) + (c, a)(b, ca) + (a, b)(c, ab)\]
holds. Prove that there exist $2011$ consecutive positive integers which are [i]amazing[/i].
[b]Note.[/b] By $(m, n)$ we denote the greatest common divisor of positive integers $m$ and $n$.
Find all pairs $(a, b)$ of positive integers such that
\[ 3^a = 2b^2 + 1. \]
Consider the expansion
\[(1 + x + x^2 + x^3 + x^4)^{496} = a_0 + a_1x + \cdots + a_{1984}x^{1984}.\]
[b](a)[/b] Determine the greatest common divisor of the coefficients $a_3, a_8, a_{13}, \ldots , a_{1983}.$
[b](b)[/b] Prove that $10^{340 }< a_{992} < 10^{347}.$
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 $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.
Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$
Two players play the following game. They alternately write divisors of
$100!$ on the blackboard, not repeating any of the numbers written before. The player after whose move the greatest common divisor of the written numbers equals $1,$ loses the game. Which player has a winning strategy?
Medians $AD$, $BE$, and $CF$ of triangle $ABC$ meet at $G$ as shown. Six small triangles, each with vertex at $G$, are formed. We draw the circles inscribed in triangles $AFG$, $BDG$, and $CDG$ as shown. Prove that if these three circles are all congruent, then $ABC$ is equilateral.
[asy]
size(200);
defaultpen(fontsize(10));
pair C=origin, B=(12,0), A=(3,14), D=midpoint(B--C), E=midpoint(A--C), F=midpoint(A--B), G=centroid(A,B,C);
draw(A--B--C--A--D^^B--E^^C--F);
draw(incircle(C,G,D)^^incircle(G,D,B)^^incircle(A,F,G));
pair point=G;
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));
label("$E$", E, dir(point--E));
label("$F$", F, dir(point--F));
label("$G$", G, dir(7));[/asy]
Prove that for every non-negative integer $m$ there exists a polynomial w with integer coefficients such that $2^m$ is the greatest common divisor of the numbers $$
a_n = 3^n + w(n), n = 0, 1, 2, ....$$
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 $a^x + x - b$.
Suppose $a$, $b$ and $c$ are integers such that the greatest common divisor of $x^2+ax+b$ and $x^2+bx+c$ is $x+1$ (in the set of polynomials in $x$ with integer coefficients), and the least common multiple of $x^2+ax+b$ and $x^2+bx+c$ $x^3-4x^2+x+6$. Find $a+b+c$.
Prove that the equation $x^3+y^3+z^3=t^4$ has infinitely many solutions in positive integers such that $\gcd(x,y,z,t)=1$.
[i]Mihai Pitticari & Sorin Rǎdulescu[/i]
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]
Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.
We say that a positive integer $n$ is lovely if there exist a positive integer $k$ and (not necessarily distinct) positive integers $d_1$, $d_2$, $\ldots$, $d_k$ such that $n = d_1d_2\cdots d_k$ and $d_i^2 \mid n + d_i$ for $i=1,2,\ldots,k$.
a) Are there infinitely many lovely numbers?
b) Is there a lovely number, greater than $1$, which is a perfect square of an integer?
Prove that for every positive integer $d > 1$ and $m$ the sequence $a_n=2^{2^n}+d$ contains two terms $a_k$ and $a_l$ ($k \neq l$) such that their greatest common divisor is greater than $m$.
[i]Proposed by T. Hakobyan[/i]
Denote by $(a, b)$ the greatest common divisor of $a$ and $b$.
Let $n$ be a positive integer such that $(n, n + 1) < (n, n + 2) <... < (n,n + 35)$. Prove that $(n, n + 35) < (n,n + 36)$.
An infinite sequence $x_1,x_2,\ldots$ of positive integers satisfies \[ x_{n+2}=\gcd(x_{n+1},x_n)+2006 \] for each positive integer $n$. Does there exist such a sequence which contains exactly $10^{2006}$ distinct numbers?