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.

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Found problems: 721

TNO 2008 Senior, 7
Tags: number theory , algebra , prime number
Find all pairs of prime numbers $p$ and $q$ such that: \[ p(p + q) = q^p+ 1. \]

2019 JBMO Shortlist, N1
Tags: number theory , prime number
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

2011 Puerto Rico Team Selection Test, 2
Tags: quadratic , number theory , prime number , algebra
Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime. Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )

2018 IFYM, Sozopol, 5
Tags: number theory , prime number
Find the solutions in prime numbers of the following equation $p^4 + q^4 + r^4 + 119 = s^2 .$

2010 Contests, 2
Tags: polynomial , ceiling function , number theory , prime number , algebra
Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2015 IFYM, Sozopol, 5
Tags: prime divisor , number theory , prime number , divisibility
Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

2018 Harvard-MIT Mathematics Tournament, 3
Tags: number theory , prime number
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?

2025 Junior Macedonian Mathematical Olympiad, 3
Tags: number theory , prime number , infinite sequence , algorithm
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.

2014 Contests, 4
Tags: modular arithmetic , number theory , prime number
The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

1977 IMO, 3
Tags: number theory , prime number , prime factorization , dirichlet s theorem
Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

PEN E Problems, 23
Tags: induction , number theory , prime number
Let $p_{1}=2, p_{2}={3}, p_{3}=5, \cdots, p_{n}$ be the first $n$ prime numbers, where $n \ge 3$. Prove that \[\frac{1}{{p_{1}}^{2}}+\frac{1}{{p_{2}}^{2}}+\cdots+\frac{1}{{p_{n}}^{2}}+\frac{1}{p_{1}p_{2}\cdots p_{n}}< \frac{1}{2}.\]

2018 AMC 12/AHSME, 5
Tags: number theory , prime number
How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

2013 Tournament of Towns, 6
Tags: prime , divide , divisor , number theory , prime number , fraction
The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

2020 International Zhautykov Olympiad, 1
Tags: number theory , zhautykov , euler s phi function , order , prime number
Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

2008 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: inequalities , abstract algebra , number theory , greatest common divisor , prime number
Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.

2025 Poland - First Round, 5
Tags: number theory , prime number
Positive integers $a, b, n$ are given. Assume that $a$ and $n$ are even, $b$ is odd and the number $ab(a+b)^{n-1}$ is divisible by $a^n+b^n$. Prove that there exist a prime number $p$, such that $p^{n+1}$ divides $a^n+b^n$.

1991 AIME Problems, 5
Tags: number theory , prime number , prime factorization
Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?

2004 Federal Competition For Advanced Students, P2, 2
Tags: prime number , number theory
Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $ \frac{1}{2004}$ is among this summands? Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$

2022 Cyprus JBMO TST, 2
Tags: number theory , prime number , prime
Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

2005 China Team Selection Test, 1
Tags: number theory , prime number , divisibility , power of 2 , prime divisor
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.

2011 All-Russian Olympiad, 3
Tags: modular arithmetic , relatively prime , prime number , number theory
For positive integers $a>b>1$, define \[x_n = \frac {a^n-1}{b^n-1}\] Find the least $d$ such that for any $a,b$, the sequence $x_n$ does not contain $d$ consecutive prime numbers. [i]V. Senderov[/i]

2020 Poland - Second Round, 5.
Tags: number theory , prime number
Let $p>$ be a prime number and $S$ be a set of $p+1$ integers. Prove that there exist pairwise distinct numbers $a_1,a_2,...,a_{p-1}\in S$ that $$ a_1+2a_2+3a_3+...+(p-1)a_{p-1}$$ is divisible by $p$.

2013 Korea - Final Round, 5
Tags: modular arithmetic , euler , prime number , number theory
Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties \[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \] Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.

2016 JBMO TST - Turkey, 7
Tags: number theory , prime number , dumb bounding
Find all pairs $(p, q)$ of prime numbers satisfying \[ p^3+7q=q^9+5p^2+18p. \]

2009 All-Russian Olympiad, 6
Tags: number theory , prime number , combinatorics
Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?