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

2013 National Olympiad First Round, 4
Tags: number theory , prime number , combinatorics
The numbers $1,2,\dots, 49$ are written on unit squares of a $7\times 7$ chessboard such that consequtive numbers are on unit squares sharing a common edge. At most how many prime numbers can a row have? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 3 $

2022 Kurschak Competition, 2
Tags: number theory , prime number , square
Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.

2022 Kazakhstan National Olympiad, 2
Tags: number theory , prime number
Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

2019 India PRMO, 21
Tags: number theory , prime number , set , prime
Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

2006 Germany Team Selection Test, 1
Tags: algebra , polynomial , number theory , composite number , prime number
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

2012 European Mathematical Cup, 2
Tags: greatest common divisor , function , prime number , number theory
Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

2012 Morocco TST, 2
Tags: number theory , prime number
Find all positive integer $n$ and prime number $p$ such that $p^2+7^n$ is a perfect square

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? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

the 11th XMO, 3
Tags: prime number , number theory
Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

2008 Romania National Olympiad, 3
Tags: inequalities , number theory , prime number
Let $ p,q,r$ be 3 prime numbers such that $ 5\leq p <q<r$. Knowing that $ 2p^2\minus{}r^2 \geq 49$ and $ 2q^2\minus{}r^2\leq 193$, find $ p,q,r$.

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

2022 Brazil Team Selection Test, 3
Tags: number theory , prime number , sequence
Let $p$ be an odd prime number and suppose that $2^h \not \equiv 1 \text{ (mod } p\text{)}$ for all integer $1 \leq h \leq p-2$. Let $a$ be an even number such that $\frac{p}{2} < a < p$. Define the sequence $a_0, a_1, a_2, \ldots$ as $$a_0 = a, \qquad a_{n+1} = p -b_n, \qquad n = 0,1,2, \ldots,$$ where $b_n$ is the greatest odd divisor of $a_n$. Show that the sequence is periodic and determine its period.

1999 Irish Math Olympiad, 2
Tags: function , number theory , prime number , algebra
A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies: $ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime; $ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$. Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$

2020 Peru IMO TST, 1
Tags: number theory , perfect power , prime number
Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2010 India IMO Training Camp, 8
Tags: prime number , number theory
Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

2023 Iran MO (3rd Round), 2
Tags: number theory , prime number
Let $N$ be the number of ordered pairs $(x,y)$ st $1 \leq x,y \leq p(p-1)$ and : $$x^{y} \equiv y^{x} \equiv 1 \pmod{p}$$ where $p$ is a fixed prime number. Show that : $$(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2$$ where $d(n)$ is the number of divisors of $n$

2020 Turkey MO (2nd round), 4
Tags: prime number , number theory
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.

2003 APMO, 3
Tags: number theory , prime number
Let $k\ge 14$ be an integer, and let $p_k$ be the largest prime number which is strictly less than $k$. You may assume that $p_k\ge 3k/4$. Let $n$ be a composite integer. Prove: (a) if $n=2p_k$, then $n$ does not divide $(n-k)!$; (b) if $n>2p_k$, then $n$ divides $(n-k)!$.

2012 JBMO TST - Macedonia, 1
Tags: prime number , number theory
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.

2021 Argentina National Olympiad, 1
Tags: number theory , prime number
Determine all pairs of prime numbers $p$ and $q$ greater than $1$ and less than $100$, such that the following five numbers: $$p+6,p+10,q+4,q+10,p+q+1,$$ are all prime numbers.

Russian TST 2014, P1
Tags: number theory , divisibility , prime number
Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$

2022 Poland - Second Round, 3
Tags: number theory , prime number , modular congruences , divisibility
Positive integers $a,b,c$ satisfying the equation $$a^3+4b+c = abc,$$ where $a \geq c$ and the number $p = a^2+2a+2$ is a prime. Prove that $p$ divides $a+2b+2$.

STEMS 2021 Math Cat B, Q2
Tags: polynomial , power of 2 , prime number , divisibility , kobayashi
Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

1982 Bulgaria National Olympiad, Problem 1
Tags: inequalities , prime number , number theory
Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$.

2007 All-Russian Olympiad Regional Round, 8.3
Tags: prime number , number theory
Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.