Found problems: 721
Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that
[list=1]
[*] $A\cap B=\{1\};$
[*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$;
[*] each prime number is a divisor of some number in $A$ and also some number in $B$;
[*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$.
[*] Each set has infinitely many composite numbers.
[/list]
For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.
The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.
Suppose an integer $x$, a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$. Find the largest value of $p$.
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.
Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.
(a) Show that there are only finitely many pairs of consecutive highly divisible
integers of the form $(a, b)$ with $a\mid b$.
(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
Find all positive integers $a$ such that $4x^2 + a$ is prime for all $x = 0, 1, \dots, a - 1$.
a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers.
b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:
1) $f$ is surjective
2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$)
3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$).
Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$
[i]Proposed by Md. Fuad Al Alam[/i]
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.
How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$.
[b]a)[/b] Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$?
[b]b)[/b] Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$?
[i]part [b]a[/b] proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$
part [b]b[/b] proposed by mostafa einollah zade[/i]
For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?
Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$
Let $p$ be a prime number. The natural numbers $m$ and $n$ are written in the system with the base $p$ as $n = a_0 + a_1p +...+ a_kp^k$ and $m = b_0 + b_1p +..+ b_kp^k$. Prove that
$${n \choose m} \equiv \prod_{i=0}^{k}{a_i \choose b_i} (mod p)$$
Let $P = \{p_1,p_2,\ldots, p_{10}\}$ be a set of $10$ different prime numbers and let $A$ be the set of all the integers greater than $1$ so that their prime decomposition only contains primes of $P$. The elements of $A$ are colored in such a way that:
[list]
[*] each element of $P$ has a different color,
[*] if $m,n \in A$, then $mn$ is the same color of $m$ or $n$,
[*] for any pair of different colors $\mathcal{R}$ and $\mathcal{S}$, there are no $j,k,m,n\in A$ (not necessarily distinct from one another), with $j,k$ colored $\mathcal{R}$ and $m,n$ colored $\mathcal{S}$, so that $j$ is a divisor of $m$ and $n$ is a divisor of $k$, simultaneously.
[/list]
Prove that there exists a prime of $P$ so that all its multiples in $A$ are the same color.
The sequence $a_n (n\geq 1)$ of natural numbers is defined as $a_{n+1}=a_n+b_n,$ where $b_n$ is the number that has the same digits as $a_n$ but in the opposite order ($b_n$ can start with $0$). For example, if $a_1=180,$ then $a_2=261, a_3=423.$
a) Decide if $a_1$ can be chosen so that $a_7$ is prime.
b) Decide if $a_1$ can be chosen so that $a_5$ is prime.
Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?
Given a positive integer \(n\), let \(\phi(n)\) denote the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). Find all possible positive integers \(k\) for which there exist positive integers \(1 \leq a_1 < a_2 < \dots < a_k\) such that:
\[
\left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024
\]