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

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

2012 Online Math Open Problems, 22
Tags: floor function , number theory , prime number
Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

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

2017 Bundeswettbewerb Mathematik, 4
Tags: number theory , sequence , recursion , prime number , integer
The sequence $a_0,a_1,a_2,\dots$ is recursively defined by \[ a_0 = 1 \quad \text{and} \quad a_n = a_{n-1} \cdot \left(4-\frac{2}{n} \right) \quad \text{for } n \geq 1. \] Prove for each integer $n \geq 1$: (a) The number $a_n$ is a positive integer. (b) Each prime $p$ with $n < p \leq 2n$ is a divisor of $a_n$. (c) If $n$ is a prime, then $a_n-2$ is divisible by $n$.

2012 European Mathematical Cup, 1
Tags: prime number , number theory
Find all positive integers $a$, $b$, $n$ and prime numbers $p$ that satisfy \[ a^{2013} + b^{2013} = p^n\text{.}\] [i]Proposed by Matija Bucić.[/i]

2007 Macedonia National Olympiad, 3
Tags: prime number , number theory
Natural numbers $a, b$ and $c$ are pairwise distinct and satisfy \[a | b+c+bc, b | c+a+ca, c | a+b+ab.\] Prove that at least one of the numbers $a, b, c$ is not prime.

2009 China Team Selection Test, 3
Tags: greatest common divisor , number theory , relatively prime , induction , prime number , prime factorization , mobius function
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2013 IFYM, Sozopol, 3
Tags: number theory , prime number
Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$

2012 National Olympiad First Round, 26
Tags: prime number , number theory
How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

2023 Brazil National Olympiad, 6
Tags: number theory , sequence , prime number
For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties: • $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$; • $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$. Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.

2011 CentroAmerican, 4
Tags: prime number , number theory
Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

1991 India Regional Mathematical Olympiad, 7
Tags: modular arithmetic , induction , prime number , number theory
Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2011 Puerto Rico Team Selection Test, 3
Tags: prime number , number theory , algebra
(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

2010 Contests, 1
Tags: arithmetic sequence , logarithm , function , prime number , number theory
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2018 USAMTS Problems, 5:
Tags: number theory , prime number
The sequence $\{a_n\}$ is defined by $a_0 = 1, a_1 = 2,$ and for $n \geq 2,$ $$a_n = a_{n-1}^2 + (a_0a_1 \dots a_{n-2})^2.$$ Let $k$ be a positive integer, and let $p$ be a prime factor of $a_k.$ Show that $p > 4(k-1).$

2017 Macedonia JBMO TST, 1
Tags: prime number , divisibility , number theory
Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

2015 Benelux, 3
Tags: number theory , prime number
Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?

2009 China Team Selection Test, 3
Tags: induction , prime factorization , prime number , relatively prime , greatest common divisor , mobius function , number theory
Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2008 Romania Team Selection Test, 2
Tags: prime number , number theory
Are there any sequences of positive integers $ 1 \leq a_{1} < a_{2} < a_{3} < \ldots$ such that for each integer $ n$, the set $ \left\{a_{k} \plus{} n\ |\ k \equal{} 1, 2, 3, \ldots\right\}$ contains finitely many prime numbers?

2023 Moldova EGMO TST, 4
Tags: prime number , system of equations , number theory , algebra
Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

2013 Harvard-MIT Mathematics Tournament, 11
Tags: number theory , prime factorization , binomial coefficient , pascal triangle , hmmt , prime number
Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

2009 AMC 12/AHSME, 19
Tags: polynomial , algebra , search , vieta , prime number , number theory
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers? $ \textbf{(A)}\ 794\qquad \textbf{(B)}\ 796\qquad \textbf{(C)}\ 798\qquad \textbf{(D)}\ 800\qquad \textbf{(E)}\ 802$

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

2021 Regional Olympiad of Mexico Southeast, 2
Tags: number theory , prime number , arithmetic sequence
Let $n\geq 2021$. Let $a_1<a_2<\cdots<a_n$ an arithmetic sequence such that $a_1>2021$ and $a_i$ is a prime number for all $1\leq i\leq n$. Prove that for all $p$ prime with $p<2021, p$ divides the diference of the arithmetic sequence.

2023 Grosman Mathematical Olympiad, 4
Tags: quadratic , prime number , number theory
Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers \[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\] to be products of two primes (not necessarily distinct).