Found problems: 1048
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
A set S consisting of $2019$ (different) positive integers has the following property:
[i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i].
What is the maximum number of prime numbers that $S$ can contain?
Let us call a positive integer $k{}$ interesting if the product of the first $k{}$ primes is divisible by $k{}$. For example the product of the first two primes is $2\cdot3 = 6$, it is divisible by 2, hence 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers?
[i]Boris Frenkin[/i]
Let $k>1$ and $n$ be natural numbers and
$p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$.
Prove that, if $p$ is prime, then $n|k!$.
p is a prime number, k is a positive integer
Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?
Let \(n\) be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.
Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$?
Let $p$ be an odd prime number. Around a circular table, $p$ students sit. We give $p$ pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the $k-$th turn we skip $2k$ students and give the next student a piece of candy. We do this until we don't give out all $p$ pieces of candy.
$a)$ How many students won't get any pieces of candy?
$b)$ How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?
Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties:
$(1)T$ consists of finitely many prime numbers;
$(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$
Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime.
[i]Lucian Petrescu[/i]
If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$
Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers
\[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \]
are integers. Prove that $p=q=r $.
[i]Nazar Agakhanov[/i]
We call a set of positive integers [i]suitable [/i] if none of its elements is coprime to the sum of all
elements of that set. Given a real number $\varepsilon \in (0,1)$, prove that, for all large enough positive
integers $N$, there exists a suitable set of size at least $\varepsilon N$, each element of which is at most $N$.
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.
Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.
Is there always a number $x$ that satisfies all the equations?
[i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]
Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.
Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.
Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.
Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\]
Proposed by Moldova
Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$.
Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?
Find all functions $f : \mathbb N \to \mathbb N$ such that:
i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$,
ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and
iii) $f(m)=1$ if and only if $m=1$.