Found problems: 721
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares
assume that A is a finite subset of prime numbers, and a is an positive integer.
prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.
Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.
Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which
primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$:
a) $x^2 +16y^2, $
b) $4x^2 +4xy+ 5y^2.$
Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.
Can the remainder of the division of a prime number $p> 30$ by $30$ be a composite?
How many triples $(p,q,n)$ are there such that $1/p+2013/q = n/5$ where $p$, $q$ are prime numbers and $n$ is a positive integer?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 4
$
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.
Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic.
[i]L. Lovasz, J. Pelikan[/i]
Determine all pairs of prime numbers $(p, q)$, with $2 \leq p, q < 100$, such that $p+6, p+10, q+4, q+10$ and $p+q+1$ are all prime numbers.
Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.
If $a$ and $b$ are positive integers, we say that $a$ [i]almost divides[/i] $b$ if $a$ divides at least one of $b - 1$ and $b + 1$. We call a positive integer $n$ [i]almost prime[/i] if the following holds: for any positive integers $a, b$ such that $n$ almost divides $ab$, we have that $n$ almost divides at least one of $a$ and $b$. Determine all almost prime numbers.
[hide = original link][url]https://mathematical.olympiad.ch/fileadmin/user_upload/Archiv/Intranet/Olympiads/Mathematics/deploy/exams/2024/FinalRound/Exam/englishFinalRound2024.pdf[/url]!![/hide]
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively.
Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes.
[i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]
Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions
a)$f(1)=1$
b)$f$ is bijective
c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.
Find all prime numbers which can be presented as a sum of two primes and difference of two primes at the same time.
For a prime number $p > 3$, define the following irreducible fraction:
$$\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1$$
Prove that $m$ is divisible by $p$.
[i]Proposed by Oleksii Masalitin[/i]
For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:
$\textbf{(A) }0\qquad
\textbf{(B) }1\qquad
\textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$
$\textbf{(D) }n-1\qquad
\textbf{(E) }n$
All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$
Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$
Find all prime numbers $p,q$ such that:
$$p^4+p^3+p^2+p=q^2+q$$
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation
\[ p_{n+2} = p_{n+1} + p_n + k \]
holds for all positive integers $n$.
Let $\mathbb{Z^+}$ and $\mathbb{P}$ denote the set of positive integers and the set of prime numbers, respectively. A set $A$ is called $S-\text{proper}$ where $A, S \subset \mathbb{Z^+}$ if there exists a positive integer $N$ such that for all $a \in A$ and for all $0 \leq b <a$ there exist $s_1, s_2, \ldots, s_n \in S$ satisfying $ b \equiv s_1+s_2+\cdots+s_n \pmod a$ and $1 \leq n \leq N.$
Find a subset $S$ of $\mathbb{Z^+}$ for which $\mathbb{P}$ is $S-\text{proper}$ but $\mathbb{Z^+}$ is not.