Found problems: 721
2016 Croatia Team Selection Test, Problem 4
Find all pairs $(p,q)$ of prime numbers such that
$$ p(p^2 - p - 1) = q(2q + 3) .$$
2024 Turkey EGMO TST, 5
Let $p$ be a given prime number. For positive integers $n,k\geq2$ let $S_1, S_2,\dots, S_n$ be unit square sets constructed by choosing exactly one unit square from each of the columns from $p\times k$ chess board. If $|S_i \cap S_j|=1$ for all $1\leq i < j \leq n$ and for any duo of unit squares which are located at different columns there exists $S_i$ such that both of these unit squares are in $S_i$ find all duos of $(n,k)$ in terms of $p$.
Note: Here we denote the number of rows by $p$ and the number of columns by $k$.
2007 Korea National Olympiad, 4
For all positive integer $ n\geq 2$, prove that product of all prime numbers less or equal than $ n$ is smaller than $ 4^{n}$.
2018 Brazil Team Selection Test, 2
Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$.
[i](R. Salimov)[/i]
2010 Poland - Second Round, 3
Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).
2016 Latvia National Olympiad, 4
In a Pythagorean triangle all sides are longer than 5. Is it possible that (a) all three sides are prime numbers, (b) exactly two sides are prime numbers. (Note: We call a triangle "Pythagorean", if it is a right-angled triangle where all sides are positive integers.)
2007 Moldova Team Selection Test, 4
Show that there are infinitely many prime numbers $p$ having the following property: there exists a natural number $n$, not dividing $p-1$, such that $p|n!+1$.
1995 AMC 12/AHSME, 29
For how many three-element sets of positive integers $\{a,b,c\}$ is it true that $a \times b \times c = 2310$?
$\textbf{(A)}\ 32 \qquad
\textbf{(B)}\ 36 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 43 \qquad
\textbf{(E)}\ 45$
2015 Caucasus Mathematical Olympiad, 1
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
2008 Bulgarian Autumn Math Competition, Problem 8.3
Prove that there exists a prime number $p$, such that the sum of digits of $p$ is a composite odd integer. Find the smallest such $p$.
2024 Macedonian Balkan MO TST, Problem 3
Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.
2020 Junior Balkаn MO, 4
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]
2024 Singapore MO Open, Q5
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$.
[i]Proposed by oneplusone[/i]
2022 Cyprus JBMO TST, 2
Determine all pairs of prime numbers $(p, q)$ which satisfy the equation
\[
p^3+q^3+1=p^2q^2
\]
2002 Korea Junior Math Olympiad, 2
Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.
2015 AIME Problems, 3
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
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?
2005 Flanders Math Olympiad, 3
Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.
2017 Macedonia JBMO TST, 1
Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.
2012 National Olympiad First Round, 26
How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$
Bangladesh Mathematical Olympiad 2020 Final, #11
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.