Found problems: 1048
Consider all triples $(x,y,p)$ of positive integers, where $p$ is a prime number, such that $4x^2 + 8y^2 + (2x-3y)p-12xy = 0$. Which below number is a perfect square number for every such triple $(x,y, p)$?
A. $4y + 1$ B. $2y + 1$ C. $8y + 1$ D. $5y - 3$ E. $8y - 1$
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$
holds.
[i](Walther Janous)[/i]
For every sequence $ (x_1, x_2, \ldots, x_n)$ of non-zero natural prime numbers, $ \{1, 2, \ldots, n\}$ arranged in any order, denote by $ f(s)$ the sum of absolute values of the differences between two consecutive members of $ s.$ Find the maximum value of $ f(s)$ where $ s$ runs through the set of all such sequences, i.e. for all sequences $ s$ with the given properties.
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.
Prove that the number
$$2^{2^k-1}-2^k-1$$is composite (not prime) for all positive integers $k>2$.
Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that
$$n+gcd(n, k)=k.$$
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]
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
There exist primes $p$ and $q$ such that
\[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633
\times 134217730 \times 2^{2202} + 1. \]
Find the remainder when $p+q$ is divided by $1000$.
[i]Proposed by Evan Chen[/i]
given $p_1,p_2,...$ be a sequence of integer and $p_1=2$,
for positive integer $n$, $p_{n+1}$ is the least prime factor of $np_1^{1!}p_2^{2!}...p_n^{n!}+1 $
prove that all primes appear in the sequence
(Proposed by Beatmania)
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
Prove that for any positive integers $x, y, z$ with $xy-z^2 = 1$ one can find non-negative integers $a, b, c, d$ such that $x = a^2 + b^2, y = c^2 + d^2, z = ac + bd$.
Set $z = (2q)!$ to deduce that for any prime number $p = 4q + 1$, $p$ can be represented as the sum of squares of two integers.
Let $p$ be a prime number greater than 5. Suppose there is an integer $k$ satisfying that $k^2+5$ is divisible by $p$. Prove that there are positive integers $m$ and $n$ such that $p^2=m^2+5n^2$
Let $p$ be a prime such that $16p+1$ is a perfect cube. A possible choice for $p$ is
[list=1]
[*] 283
[*] 307
[*] 593
[*] 691
[/list]
$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$.
$a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite.
$b)$ Find 3 different prime numbers that do not divide any terms of this sequence.
Determine all pairs of prime numbers $(p, q)$ which satisfy the equation
\[
p^3+q^3+1=p^2q^2
\]
Find all prime numbers $p$ and $q$ such that
$$1 + \frac{p^q - q^p}{p + q}$$
is a prime number.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite.
Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$.
Tuymaada Olympiad 2004, C.A.Grimm. USA
Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?
Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$.
Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.
Find all prime numbers of the form $10101...01$.
Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions:
$$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.
Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.