This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 171

2021 CHKMO, 2
Tags: number theory , primes , prime factorization
For each positive integer $n$ larger than $1$ with prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, its [i]signature[/i] is defined as the sum $\alpha_1+\alpha_2+\cdots+\alpha_k$. Does there exist $2020$ consecutive positive integers such that among them, there are exactly $1812$ integers whose signatures are strictly smaller than $11$?

2009 Korea Junior Math Olympiad, 1
Tags: number theory , multiple , Product , primes
For primes $a, b,c$ that satis fy the following, calculate $abc$. $\bullet$ $b + 8$ is a multiple of $a$, $\bullet$ $b^2 - 1$ is a multiple of $a$ and $c$ $\bullet$ $b + c = a^2 - 1$.

2020 Dutch BxMO TST, 5
Tags: number theory , primes , Product
A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

2013 NZMOC Camp Selection Problems, 2
Tags: number theory , primes , prime , Difference
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).

2015 ELMO Problems, 4
Tags: Elmo , number theory , primes
Let $a > 1$ be a positive integer. Prove that for some nonnegative integer $n$, the number $2^{2^n}+a$ is not prime. [i]Proposed by Jack Gurev[/i]

2022 239 Open Mathematical Olympiad, 5
Tags: number theory , primes
Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

2018 Saudi Arabia IMO TST, 1
Tags: prime divisors , Divisors , primes , theory
Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

Oliforum Contest V 2017, 4
Tags: number theory , primes
Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and de fine $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

1994 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine , primes , number theory , Diophantine equation
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2010 Philippine MO, 1
Tags: number theory , primes , PMO , 2010
Find all primes that can be written both as a sum of two primes and as a difference of two primes.

2022 239 Open Mathematical Olympiad, 4
Tags: number theory , primes , Natural Numbers
Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.

2000 Tuymaada Olympiad, 5
Tags: number theory , primes , divisible
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

Mathematical Minds 2024, P6
Tags: Sequences , recursion , primes , number theory
Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

2015 Chile National Olympiad, 2
Tags: number theory , primes
Find all prime numbers that do not have a multiple ending in $2015$.

2010 QEDMO 7th, 1
Tags: number theory , primes
Find all natural numbers $n$ for which both $n^n + 1$ and $(2n)^{2n} + 1$ are prime numbers.

2010 Estonia Team Selection Test, 1
Tags: number theory , GCD , primes , power of prime , coprime
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

2013 Hanoi Open Mathematics Competitions, 1
Tags: primes , minimum , number theory
Write $2013$ as a sum of $m$ prime numbers. The smallest value of $m$ is: (A): $2$, (B): $3$, (C): $4$, (D): $1$, (E): None of the above.

Oliforum Contest V 2017, 3
Tags: number theory , Product , primes
Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$ (Paolo Leonetti)

2011 Silk Road, 4
Tags: number theory , primes , representation
Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

2022 Serbia National Math Olympiad, P6
Tags: algebra , primes , Serbian competition
Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$, divisible by $p$ or $q$.

VII Soros Olympiad 2000 - 01, 9.3
Tags: number theory , Sum , primes
Write $102$ as the sum of the largest number of distinct primes.

2005 Korea Junior Math Olympiad, 6
Tags: number theory , primes , set theory
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.

2010 Thailand Mathematical Olympiad, 6
Tags: primes , number theory , Divide
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

1995 India National Olympiad, 6
Tags: number theory , greatest common divisor , number theory solved , primes
Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

2018 Malaysia National Olympiad, A6
Tags: number theory , primes
A [i]semiprime [/i] is a positive integer that is a product of two prime numbers. For example, $9$ and $10$ are semiprimes. How many semiprimes less than $100$ are there?