Found problems: 171
1965 Poland - Second Round, 4
Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.
2016 Mathematical Talent Reward Programme, MCQ: P 4
There are 168 primes below 1000. Then sum of all primes below 1000 is
[list=1]
[*] 11555
[*] 76127
[*] 57298
[*] 81722
[/list]
2022 AMC 10, 13
The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes?
$\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$
2017 Puerto Rico Team Selection Test, 5
Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.
2017 Romania Team Selection Test, P4
Given a positive odd integer $n$, show that the arithmetic mean of fractional parts $\{\frac{k^{2n}}{p}\}, k=1,..., \frac{p-1}{2}$ is the same for infinitely many primes $p$ .
1999 Mexico National Olympiad, 2
Prove that there are no $1999$ primes in an arithmetic progression that are all less than $12345$.
2011 Belarus Team Selection Test, 1
Given natural number $a>1$ and different odd prime numbers $p_1,p_2,...,p_n$ with
$a^{p_1}\equiv 1$ (mod $p_2$), $a^{p_2}\equiv 1$ (mod $p_3$), ..., $a^{p_n}\equiv 1$(mod $p_1$).
Prove that
a) $(a-1)\vdots p_i$ for some $i=1,..,n$
b) Can $(a-1)$ be divisible by $p_i $for exactly one $i$ of $i=1,...,n$?
I. Bliznets
1997 Czech And Slovak Olympiad IIIA, 4
Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.
2014 IFYM, Sozopol, 2
The radius $r$ of a circle with center at the origin is an odd integer.
There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers.
Determine $r$.
2015 India IMO Training Camp, 2
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
2020 New Zealand MO, 4
Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.
1989 Austrian-Polish Competition, 3
Find all natural numbers $N$ (in decimal system) with the following properties:
(i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes,
(ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.
2024 Kosovo Team Selection Test, P1
Find all prime numbers $p$ and $q$ such that $p^q + 5q - 2$ is also a prime number.
2008 Singapore Junior Math Olympiad, 4
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?
2002 VJIMC, Problem 2
Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.
2023 Thailand TST, 1
Find all positive integers $n>2$ such that
$$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$
2016 Latvia Baltic Way TST, 17
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?
1994 All-Russian Olympiad Regional Round, 10.5
Find all primes that can be written both as a sum and as a difference of two primes (note that $ 1$ is not a prime).
2014 IMO Shortlist, N5
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]
2014 Junior Balkan Team Selection Tests - Romania, 3
Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.
1954 Moscow Mathematical Olympiad, 276
a) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31$ (the product of primes $2$ to $31$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.
b) Let $1, 2, 3, 5, 6, 7, 10, .., N$ be all the divisors of $N = 2\cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ (the product of primes $2$ to $37$) written in increasing order. Below this series of divisors, write the following series of $1$’s or $-1$’s: write $1$ below any number that factors into an even number of prime factors and below a $1$, write $-1$ below the remaining numbers. Prove that the sum of the series of $1$’s and $-1$’s is equal to $0$.
1998 Switzerland Team Selection Test, 6
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.
2015 Saudi Arabia BMO TST, 4
Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$.
Malik Talbi
1996 Estonia National Olympiad, 4
Prove that for each prime number $p > 5$ there exists a positive integer n such that $p^n$ ends in $001$ in decimal representation.
2015 Brazil Team Selection Test, 2
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$.
[i]Proposed by Belgium[/i]