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

1987 IMO, 3
Tags: algebra , polynomial , number theory , prime number
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

1992 Spain Mathematical Olympiad, 4
Tags: number theory , prime number , arithmetic sequence
Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.

2012 Stanford Mathematics Tournament, 1
Tags: number theory , prime number
Define a number to be $boring$ if all the digits of the number are the same. How many positive integers less than $10000$ are both prime and boring?

2015 Greece National Olympiad, 1
Tags: diophantine equation , number theory , prime number
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

2025 Francophone Mathematical Olympiad, 4
Tags: number theory , prime number
Determine all sequences of strictly positive integers $a_1, a_2, a_3, \ldots$ satisfying the following two conditions: [list] [*]There exists an integer $M > 0$ such that, for all indices $n \geqslant 1$, $0 < a_n \leqslant M$. [*]For any prime number $p$ and for any index $n \geqslant 1$, the number \[ a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p} \] is a multiple of $p$. [/list]

2017 Dutch IMO TST, 2
Tags: number theory , prime number
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

1977 IMO Shortlist, 10
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.)

2015 Nordic, 2
Tags: prime number , number theory
Find the primes ${p, q, r}$, given that one of the numbers ${pqr}$ and ${p + q + r}$ is ${101}$ times the other.

1990 IMO Longlists, 98
Tags: number theory , decimal representation , prime number , digit
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

1998 IMO, 6
Tags: number theory , prime number , algebra , functional equation
Determine the least possible value of $f(1998),$ where $f:\Bbb{N}\to \Bbb{N}$ is a function such that for all $m,n\in {\Bbb N}$, \[f\left( n^{2}f(m)\right) =m\left( f(n)\right) ^{2}. \]

2017 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , prime number
Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.

2005 AMC 12/AHSME, 18
Tags: number theory , prime number
Call a number "prime-looking" if it is composite but not divisible by 2, 3, or 5. The three smallest prime-looking numbers are 49, 77, and 91. There are 168 prime numbers less than 1000. How many prime-looking numbers are there less than 1000? $ \textbf{(A)}\ 100 \qquad \textbf{(B)}\ 102 \qquad \textbf{(C)}\ 104 \qquad \textbf{(D)}\ 106 \qquad \textbf{(E)}\ 108$

2025 6th Memorial "Aleksandar Blazhevski-Cane", P1
Tags: number theory , prime number , diophantine equation
Determine all triples of prime numbers $(p, q, r)$ that satisfy \[p2^q + r^2 = 2025.\] Proposed by [i]Ilija Jovcevski[/i]

2010 Slovenia National Olympiad, 1
Tags: prime number , number theory
Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.

2015 India PRMO, 15
Tags: number theory , digit , prime number
$15.$ Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers, $x,y,$ and $10x+y,$ where $x$ and $y$ are digits. What is the sum of digits of $n ?$

2019 IOM, 1
Tags: number theory , prime number
Three prime numbers $p,q,r$ and a positive integer $n$ are given such that the numbers \[ \frac{p+n}{qr}, \frac{q+n}{rp}, \frac{r+n}{pq} \] are integers. Prove that $p=q=r $. [i]Nazar Agakhanov[/i]

2024 Czech and Slovak Olympiad III A, 6
Tags: geometry , congruent triangles , side bash , prime number
Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.

1996 Estonia National Olympiad, 4
Tags: number theory , prime number
Can the remainder of the division of a prime number $p> 30$ by $30$ be a composite?

2023 VN Math Olympiad For High School Students, Problem 3
Tags: algebra , number theory , prime number
Given a polynomial with integer coefficents with degree $n>0:$$$P(x)=a_nx^n+...+a_1x+a_0.$$ Assume that there exists a prime number $p$ satisfying these conditions: [i]i)[/i] $p|a_i$ for all $0\le i<n,$ [i]ii)[/i] $p\nmid a_n,$ [i]iii)[/i] $p^2\nmid a_0.$ Prove that $P(x)$ is irreducible in $\mathbb{Z}[x].$

2021 Romania National Olympiad, 3
Tags: number theory , prime number
Let $n\ge 2$ be a positive integer such that the set of $n$th roots of unity has less than $2^{\lfloor\sqrt n\rfloor}-1$ subsets with the sum $0$. Show that $n$ is a prime number. [i]Cristi Săvescu[/i]

2017 Israel Oral Olympiad, 3
Tags: number theory , prime , prime number
2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2019 Olympic Revenge, 2
Tags: perfect square , number theory , prime number , my 100th post
Prove that there exist infinitely many positive integers $n$ such that the greatest prime divisor of $n^2+1$ is less than $n \cdot \pi^{-2019}.$

2019 Pan-African Shortlist, N2
Tags: number theory , prime number , sums and products
Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

1998 AMC 12/AHSME, 12
Tags: number theory , prime number , logarithm
How many different prime numbers are factors of $ N$ if \[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11? \]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$

2010 Poland - Second Round, 3
Tags: inequalities , prime number , number theory
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. $p_i|k+i$).