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: 715

2003 Pan African, 3
Tags: number theory , prime numbers
Does there exists a base in which the numbers of the form: \[ 10101, 101010101, 1010101010101,\cdots \] are all prime numbers?

2011 China Western Mathematical Olympiad, 1
Tags: modular arithmetic , number theory , prime numbers , number theory unsolved
Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?

2025 Junior Macedonian Mathematical Olympiad, 3
Tags: number theory , prime numbers , infinite sequence , Junior , JMMO , JBMO TST , 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.

2000 Flanders Math Olympiad, 3
Tags: number theory , prime numbers
Let $p_n$ be the $n$-th prime. ($p_1=2$) Define the sequence $(f_j)$ as follows: - $f_1=1, f_2=2$ - $\forall j\ge 2$: if $f_j = kp_n$ for $k<p_n$ then $f_{j+1}=(k+1)p_n$ - $\forall j\ge 2$: if $f_j = p_n^2$ then $f_{j+1}=p_{n+1}$ (a) Show that all $f_i$ are different (b) from which index onwards are all $f_i$ at least 3 digits? (c) which integers do not appear in the sequence? (d) how many numbers with less than 3 digits appear in the sequence?

2020 Latvia Baltic Way TST, 15
Tags: number theory , prime numbers
Let $p$ be a prime. Prove that $p^2+p+1$ is never a perfect cube.

2020 MMATHS, I3
Tags: number theory , prime numbers
Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$. [i]Proposed by Andrew Wu[/i]

1974 IMO Longlists, 35
Tags: number theory , prime numbers , number theory proposed
If $p$ and $q$ are distinct prime numbers, then there are integers $x_0$ and $y_0$ such that $1 = px_0 + qy_0.$ Determine the maximum value of $b - a$, where $a$ and $b$ are positive integers with the following property: If $a \leq t \leq b$, and $t$ is an integer, then there are integers $x$ and $y$ with $0 \leq x \leq q - 1$ and $0 \leq y \leq p - 1$ such that $t = px + qy.$

1995 Moldova Team Selection Test, 2
Tags: number theory , prime numbers
Let $p{}$ be a prime number. Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions if and only if the equation $y^2-y+25-pt=0$ with $y,t\in\mathbb{Z}$ has solutions.

1996 Iran MO (3rd Round), 4
Tags: number theory , prime numbers , number theory proposed , combinatorics
Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that \[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]

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

2013 All-Russian Olympiad, 3
Tags: quadratics , modular arithmetic , number theory , prime numbers , number theory proposed
Find all positive integers $k$ such that for the first $k$ prime numbers $2, 3, \ldots, p_k$ there exist positive integers $a$ and $n>1$, such that $2\cdot 3\cdot\ldots\cdot p_k - 1=a^n$. [i]V. Senderov[/i]

2021 Azerbaijan EGMO TST, 1
Tags: prime numbers , prime , number theory , AZE EGMO TST
p is a prime number, k is a positive integer Find all (p, k): $k!=(p^3-1)(p^3-p)(p^3-p^2)$

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

2021 Romania National Olympiad, 3
Tags: number theory , prime numbers
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]

2010 Canadian Mathematical Olympiad Qualification Repechage, 4
Tags: number theory , prime numbers , number theory proposed
Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2007 Silk Road, 1
Tags: number theory , prime numbers
On the board are written $2 , 3 , 5 ,... , 2003$ , that is, all the prime numbers of the interval $[2,2007]$ . The operation of [i]simplification [/i] is the replacement of two numbers $a , b$ by a maximal prime number not exceeding $\sqrt{a^2-a b+b^2}$ . First, the student erases the number $q, 2<q<2003$, then applies the [i]simplification [/i] operation to the remaining numbers until one number remains. Find the maximum possible and minimum possible values of the number obtained in the end. How do these values depend on the number $q$?

2019 Korea Junior Math Olympiad., 5
Tags: KJMO , number theory , prime numbers , algebra , polynomial
For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.

1960 AMC 12/AHSME, 33
Tags: number theory , prime numbers , AMC
You are given a sequence of $58$ terms; each term has the form $P+n$ where $P$ stands for the product $2 \times 3 \times 5 \times... \times 61$ of all prime numbers less than or equal to $61$, and $n$ takes, successively, the values $2, 3, 4, ...., 59$. let $N$ be the number of primes appearing in this sequence. Then $N$ is: $ \textbf{(A)}\ 0\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 17\qquad\textbf{(D)}\ 57\qquad\textbf{(E)}\ 58 $

2019 IOM, 1
Tags: number theory , prime numbers , Kvant
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]

2022 IFYM, Sozopol, 3
Tags: number theory , prime numbers
Let $p_1,p_2,\dots ,p_n$ be all prime numbers lesser than $2^{100}$. Prove that $\frac{1}{p_1} +\frac{1}{p_2} +\dots +\frac{1}{p_n} <10$.

Bangladesh Mathematical Olympiad 2020 Final, #11
Tags: prime numbers , number theory
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.

1994 China Team Selection Test, 2
Tags: algebra , polynomial , calculus , integration , Gauss , number theory , prime numbers
Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1973 Miklós Schweitzer, 3
Tags: logarithms , floor function , number theory , prime numbers , number theory proposed
Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

2024 Mozambican National MO Selection Test, P3
Tags: nt , number theory , Diophantine equation , factorization , positive integers , prime numbers
Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

2017 Romania National Olympiad, 4
Tags: number theory , prime , Digits , prime numbers
Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.