Found problems: 721
A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?
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]
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
Use each of the digits 1,2,3,4,5,6,7,8,9 exactly twice to form distinct prime numbers whose sum is as small as possible. What must this minimal sum be? (Note: The five smallest primes are 2,3,5,7, and 11)
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$
Assume that there exists a prime number $p$ satisfying these conditions:
[i]i)[/i] $p|a_i$ for all $0\le i<n,$
[i]ii)[/i] $p\nmid a_n,$
[i]iii)[/i] $p^2\nmid a_0.$
Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$
Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$.
[i]Proposed by oneplusone[/i]
Let $n$ be a positive integer such that $n + 1$ is divisible by 24. Prove that the sum of all the positive divisors of $n$ is divisible by 24.
Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$).
Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.
Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.
Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.
Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that
$$p \mid m,n$$
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$?
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$
Prove that, for all integers $a, b$, there exists a positive integer $n$, such that the number $n^2+an+b$ has at least $2018$ different prime divisors.
Find all pairs of natural numbers n and prime numbers p such that $\sqrt{n+\frac{p}{n}}$ is a natural number.
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
Let $a,b$ and $c$ be positive integers such that $a!+b+c,b!+c+a$ and $c!+a+b$ are prime numbers. Show that $\frac{a+b+c+1}{2}$ is also a prime number.
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.
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.
Let $ n$ be the largest integer that is the product of exactly $ 3$ distinct prime numbers, $ d$, $ e$, and $ 10d\plus{}e$, where $ d$ and $ e$ are single digits. What is the sum of the digits of $ n$?
$ \textbf{(A)}\ 12 \qquad
\textbf{(B)}\ 15 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 21 \qquad
\textbf{(E)}\ 24$
Find all triples $(m,p,q)$ such that
\begin{align*}
2^mp^2 +1=q^7,
\end{align*}
where $p$ and $q$ are ptimes and $m$ is a positive integer.