Found problems: 721
2016 Croatia Team Selection Test, Problem 4
Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime.
Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.
2015 AIME Problems, 3
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.
2016 Iran MO (3rd Round), 1
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:
$$q |(x+1)^p-x^p$$
If and only if $$q \equiv 1 \pmod p$$
2013 Dutch IMO TST, 3
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.
1987 Czech and Slovak Olympiad III A, 4
Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$
2023 Bundeswettbewerb Mathematik, 1
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$.
2009 China Team Selection Test, 2
Find all integers $ n\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$), there exists an integer polynomial $ f(x)$ such that congruence equation $ f(x)\equiv 0 (mod n)$ exactly has $ k$ roots $ x\equiv a_{1},a_{2},\cdots,a_{k} (mod n).$
2022 IMO, 3
Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.
2023 Durer Math Competition Finals, 1
Nüx has three moira daughters, whose ages are three distinct prime numbers, and the sum of their squares is also a prime number. What is the age of the youngest moira?
2016 Iran Team Selection Test, 3
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.
2015 IFYM, Sozopol, 1
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.
2003 Pan African, 3
Does there exists a base in which the numbers of the form:
\[ 10101, 101010101, 1010101010101,\cdots \]
are all prime numbers?
2014 Saudi Arabia BMO TST, 1
A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.
2023 Stars of Mathematics, 1
Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.
2017 Pan-African Shortlist, N2
For which prime numbers $p$ can we find three positive integers $n$, $x$ and $y$ such that $p^n = x^3 + y^3$?
2015 JBMO Shortlist, NT5
Check if there exists positive integers $ a, b$ and prime number $p$ such that $a^3-b^3=4p^2$
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]
2020 Hong Kong TST, 4
Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.
1999 Singapore Team Selection Test, 3
Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$
1998 National Olympiad First Round, 18
Let $ p_{1} <p_{2} <\ldots <p_{24}$ be the prime numbers on the interval $ \left[3,100\right]$. Find the smallest value of $ a\ge 0$ such that $ \sum _{i\equal{}1}^{24}p_{i}^{99!} \equiv a\, \, \left(mod\, 100\right)$.
$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 48 \qquad\textbf{(D)}\ 50 \qquad\textbf{(E)}\ 99$
2004 Regional Competition For Advanced Students, 4
The sequence $ < x_n >$ is defined through:
$ x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1$ for $ n > 0$
Let $ x_1$ be a non-negative integer smaller than $ 204$ so that all members of the sequence are non-negative integers.
Show that there exist infinitely many prime numbers in this sequence.
2007 May Olympiad, 4
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins.
Which player has the winning strategy?
2004 Baltic Way, 8
Let $f\left(x\right)$ be a non-constant polynomial with integer coefficients, and let $u$ be an arbitrary positive integer. Prove that there is an integer $n$ such that $f\left(n\right)$ has at least $u$ distinct prime factors and $f\left(n\right) \neq 0$.
2020 Iran RMM TST, 1
For all prime $p>3$ with reminder $1$ or $3$ modulo $8$ prove that the number triples $(a,b,c), p=a^2+bc, 0<b<c<\sqrt{p}$ is odd.
[i]Proposed by Navid Safaie[/i]
2011 ISI B.Stat Entrance Exam, 7
[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$.
[b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.