Found problems: 721
2021 Centroamerican and Caribbean Math Olympiad, 1
An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.
2014 USAMTS Problems, 2:
Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.
2017 China Northern MO, 8
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:
(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$, wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).
(2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$.
(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.
Assume that all wolves are very smart, then how many wolves will remain in the end?
2016 JBMO TST - Turkey, 7
Find all pairs $(p, q)$ of prime numbers satisfying
\[ p^3+7q=q^9+5p^2+18p. \]
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2017 Iran MO (2nd Round), 1
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$
b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
2012 Mathcenter Contest + Longlist, 10
The table size $8 \times 8$ contains the numbers $1,2,...,8$ in each amount as much as you want provided that two numbers that are adjacent vertically, horizontally, diagonally are relative primes. Prove that some number appears in the table at least $12$ times.
[i](PP-nine)[/i]
2011 Danube Mathematical Competition, 3
Determine all positive integer numbers $n$ satisfying the following condition:
the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.
1981 Czech and Slovak Olympiad III A, 4
Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.
2005 China Team Selection Test, 1
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.
Russian TST 2015, P2
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 $p{}$.
2008 Bosnia And Herzegovina - Regional Olympiad, 3
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.
2015 Romania Masters in Mathematics, 5
Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$
2011 Pre-Preparation Course Examination, 5
suppose that $v(x)=\sum_{p\le x,p\in \mathbb P}log(p)$ (here $\mathbb P$ denotes the set of all positive prime numbers). prove that the two statements below are equivalent:
[b]a)[/b] $v(x) \sim x$ when $x \longrightarrow \infty$
[b]b)[/b] $\pi (x) \sim \frac{x}{ln(x)}$ when $x \longrightarrow \infty$. (here $\pi (x)$ is number of the prime numbers less than or equal to $x$).
2019 Olympic Revenge, 5
Define $f: \mathbb{N} \rightarrow \mathbb{N}$ by $$f(n) = \sum \frac{(1+\sum_{i=1}^{n} t_i)!}{(1+t_1) \cdot \prod_{i=1}^{n} (t_i!) }$$
where the sum runs through all $n$-tuples such that $\sum_{j=1}^{n}j \cdot t_j=n$ and $t_j \ge 0$ for all $1 \le j \le n$.
Given a prime $p$ greater than $3$, prove that $$\sum_{1 \le i < j <k \le p-1 } \frac{f(i)}{i \cdot j \cdot k} \equiv \sum_{1 \le i < j <k \le p-1 } \frac{2^i}{i \cdot j \cdot k} \pmod{p}.$$
2000 Saint Petersburg Mathematical Olympiad, 10.7
We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number?
[I]Proposed by S. Berlov, S. Ivanov[/i]
2017 Iran Team Selection Test, 4
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]
2018 Bosnia and Herzegovina Junior BMO TST, 2
Find all integer triples $(p,m,n)$ that satisfy:
$p^m-n^3=27$ where $p$ is a prime number.
2021 Iran Team Selection Test, 4
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have :
$$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$
Who has the winning strategy.
Proposed by [i]Alireza Haghi[/i]
2007 Tournament Of Towns, 5
Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.
2010 Singapore MO Open, 3
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$