Found problems: 715
2015 Greece National Olympiad, 1
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$
1987 Brazil National Olympiad, 1
$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.
2018 Stars of Mathematics, 2
Find the smallest natural $ k $ such that among any $ k $ distinct and pairwise coprime naturals smaller than $ 2018, $ a prime can be found.
[i]Vlad Robu[/i]
2018 Bosnia And Herzegovina - Regional Olympiad, 3
Let $p$ and $q$ be prime numbers such that $p^2+pq+q^2$ is perfect square. Prove that $p^2-pq+q^2$ is prime
2018 AMC 10, 11
Which of the following expressions is never a prime number when $p$ is a prime number?
$\textbf{(A) } p^2+16 \qquad \textbf{(B) } p^2+24 \qquad \textbf{(C) } p^2+26 \qquad \textbf{(D) } p^2+46 \qquad \textbf{(E) } p^2+96$
2015 Brazil Team Selection Test, 3
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]
2020 JBMO Shortlist, 8
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]
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]
2025 Poland - First Round, 3
Let $n$ be a product of 2024 different prime numbers. Find the number of positive integers $k$, such that
$$n+gcd(n, k)=k.$$
2022 Romania Team Selection Test, 3
Consider a prime number $p\geqslant 11$. We call a triple $a,b,c$ of natural numbers [i]suitable[/i] if they give non-zero, pairwise distinct residues modulo $p{}$. Further, for any natural numbers $a,b,c,k$ we define \[f_k(a,b,c)=a(b-c)^{p-k}+b(c-a)^{p-k}+c(a-b)^{p-k}.\]Prove that there exist suitable $a,b,c$ for which $p\mid f_2(a,b,c)$. Furthermore, for each such triple, prove that there exists $k\geqslant 3$ for which $p\nmid f_k(a,b,c)$ and determine the minimal $k{}$ with this property.
[i]Călin Popescu and Marian Andronache[/i]
2019 Moroccan TST, 4
Let $p$ be a prime number. Find all the positive integers $n$ such that $p+n$ divides $pn$
1984 All Soviet Union Mathematical Olympiad, 386
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
2016 Mathematical Talent Reward Programme, MCQ: P 8
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]
2004 Finnish National High School Mathematics Competition, 4
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.
Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
2011 Greece Team Selection Test, 1
Find all prime numbers $p,q$ such that:
$$p^4+p^3+p^2+p=q^2+q$$
2020 OMpD, 3
Determine all integers $n$ such that both of the numbers:
$$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$
are both prime numbers.
2016 Hong Kong TST, 1
Find all prime numbers $p$ and $q$ such that $p^2|q^3+1$ and $q^2|p^6-1$
2018 All-Russian Olympiad, 1
Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.
2020 Turkey MO (2nd round), 4
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 $2p^2+2p+1$.
2018 AMC 10, 5
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number?
$\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$
2018 Harvard-MIT Mathematics Tournament, 3
There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?
2023 Romania EGMO TST, P2
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.
1993 All-Russian Olympiad, 1
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
2012 BMT Spring, 7
Let $ a $ , $ b $ , $ c $ , $ d $ , $ (a + b + c + 18 + d) $ , $ (a + b + c + 18 - d) $ , $ (b + c) $ , and $ (c + d) $ be distinct prime numbers such that $ a + b + c = 2010 $, $ a $, $ b $, $ c $, $ d \neq 3 $ , and $ d \le 50 $. Find the maximum value of the difference between two of these prime numbers.
2022 MMATHS, 10
Define a function $f$ on the positive integers as follows: $f(n) = m$, where $m$ is the least positive integer such that $n$ is a factor of $m^2$. Find the smallest integer $M$ such that $\sqrt{M}$ is both a product of prime numbers, of which there are at least $3$, and a factor of $$\sum_{ d|M} f(d),$$ the sum of $f(d)$ for all positive integers $d$ that divide $M$.