Found problems: 526
Let $a$ be a positive integer. We say that a positive integer $b$ is [i]$a$-good[/i] if $\tbinom{an}{b}-1$ is divisible by $an+1$ for all positive integers $n$ with $an \geq b$. Suppose $b$ is a positive integer such that $b$ is $a$-good, but $b+2$ is not $a$-good. Prove that $b+1$ is prime.
Find all permutations $(a_1, a_2,...,a_{2021})$ of $(1,2,...,2021)$, such that for every two positive integers $m$ and $n$ with difference bigger than $20^{21}$, the following inequality holds:
$GCD(m+1, n+a_1)+GCD(m+2, n+a_2)+...+GCD(m+2021, n+a_{2021})<2|m-n|$.
Let $A$ be the set of all integers $n$ such that $1 \le n \le 2021$ and $\text{gcd}(n,2021)=1$. For every nonnegative integer $j$, let
\[
S(j)=\sum_{n \in A}n^j.
\]
Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.
For each positive integer $n$, denote by $\omega(n)$ the number of distinct prime divisors of $n$ (for example, $\omega(1)=0$ and $\omega(12)=2$). Find all polynomials $P(x)$ with integer coefficients, such that whenever $n$ is a positive integer satisfying $\omega(n)>2023^{2023}$, then $P(n)$ is also a positive integer with
\[\omega(n)\ge\omega(P(n)).\]
Greece (Minos Margaritis - Iasonas Prodromidis)
For positive integer $a \geq 2$, denote $N_a$ as the number of positive integer $k$ with the following property: the sum of squares of digits of $k$ in base a representation equals $k$. Prove that:
a.) $N_a$ is odd;
b.) For every positive integer $M$, there exist a positive integer $a \geq 2$ such that $N_a \geq M$.
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.
Let $a$, $b$, and $n$ be positive integers with $\gcd (a, b)=1$. Without using Dirichlet's theorem, show that there are infinitely many $k \in \mathbb{N}$ such that $\gcd(ak+b, n)=1$.
Find all prime numbers that do not have a multiple ending in $2015$.
Let $a, b, c$ be positive integers such that $a$ and $b$ are relatively prime and $c$ is relatively prime either to $a$ or $b$. Prove that there exist infinitely many triples $(x, y, z)$ of distinct positive integers such that \[x^{a}+y^{b}= z^{c}.\]
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 $m>1$ be an integer. Find the smallest positive integer $n$, such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:
i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime
ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$
Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the remainder when $S$ is divided by $1000$.
$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.
Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$.
a) Find all $n$ such that $A_{n}\neq \emptyset$
b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero.
c) Is there $n$ such that $|{A_{n}}| = 130$?
A lattice point $X$ in the plane is said to be [i]visible[/i] from the origin $O$ if the line segment $OX$ does not contain any other lattice points. Show that for any positive integer $n$, there is square $ABCD$ of area $n^{2}$ such that none of the lattice points inside the square is visible from the origin.
A number $n$ is [i]interesting[/i] if 2018 divides $d(n)$ (the number of positive divisors of $n$). Determine all positive integers $k$ such that there exists an infinite arithmetic progression with common difference $k$ whose terms are all interesting.
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$.
[i]Proposed by Evan Chen[/i]
For a polynomial $P$ and a positive integer $n$, define $P_n$ as the number of positive integer pairs $(a,b)$ such that $a<b \leq n$ and $|P(a)|-|P(b)|$ is divisible by $n$. Determine all polynomial $P$ with integer coefficients such that $P_n \leq 2021$ for all positive integers $n$.
Find the remainder when $19^{92}$ is divided by 92.
Consider the sequence in which $a_1 = 1$ and $a_n$ is obtained by juxtaposing the decimal representation of $n$ at the end of the decimal representation of $a_{n-1}$. That is, $a_1 = 1$, $a_2 = 12$, $a_3 = 123$, $\dots$ , $a_9 = 123456789$, $a_{10} = 12345678910$ and so on. Prove that infinitely many numbers of this sequence are multiples of $7$.
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$.
(1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$.
(2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.
Compute the last two digits of $$9^{2020} + 9^{2020^2}+ ... + 9^{2020^{2020}}$$