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

2018 IFYM, Sozopol, 1
Tags: prime number , number theory
Find all prime numbers $p$ and all positive integers $n$, such that $n^8 - n^2 = p^5 + p^2$

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 $a+2b+2$.

2021 Bangladeshi National Mathematical Olympiad, 1
Tags: number theory , prime number , counting
How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

2014 Balkan MO Shortlist, N2
Tags: number theory , prime number
$\boxed{N2}$ Let $p$ be a prime numbers and $x_1,x_2,...,x_n$ be integers.Show that if \[x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}\] for all positive integers n then $x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.$

2013 Bangladesh Mathematical Olympiad, 7
Tags: number theory , prime number
Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

2018 Korea Winter Program Practice Test, 4
Tags: prime number , number theory
Let $p=4k+1$ be a prime. $S$ is a set of all possible residues equal or smaller then $2k$ when $\frac{1}{2} \binom{2k}{k} n^k$ is divided by $p$. Show that \[ \sum_{x \in S} x^2 =p \]

1997 Mexico National Olympiad, 1
Tags: prime number , prime , number theory
Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

2019 Centers of Excellency of Suceava, 1
Tags: number theory , prime number
Prove that if a prime is the sum of four perfect squares then the product of two of these is equal to the product of the other two. [i]Gherghe Stoica[/i]

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

2016 India IMO Training Camp, 3
Tags: number theory , set , prime number
Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

2011 All-Russian Olympiad, 3
Tags: relatively prime , modular arithmetic , 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]

2017 Germany, Landesrunde - Grade 11/12, 3
Tags: number theory , prime number , exponent , power , catalan
Find the smallest prime number that can not be written in the form $\left| 2^a-3^b \right|$ with non-negative integers $a,b$.

2023 Brazil Team Selection Test, 4
Tags: number theory , prime number , multiple
Find all positive integers $n$ with the following property: There are only a finite number of positive multiples of $n$ that have exactly $n$ positive divisors.

2022 Brazil Team Selection Test, 3
Tags: prime number , sequence , number theory
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.

Kvant 2020, M2597
Tags: prime number , number theory
Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]

2010 Bosnia Herzegovina Team Selection Test, 1
Tags: prime number , number theory
$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

2015 Thailand TSTST, 2
Tags: prime number , number theory
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.

Kvant 2021, M2636
Tags: number theory , prime number
We call a natural number $p{}$ [i]simple[/i] if for any natural number $k{}$ such that $2\leqslant k\leqslant \sqrt{p}$ the inequality $\{p/k\}\geqslant 0,01$ holds. Is the set of simple prime numbers finite? [i]Proposed by M. Didin[/i]

2014 Junior Regional Olympiad - FBH, 4
Tags: number theory , prime number
Find all prime numbers $p$ and $q$ such that $3p^2q+2pq^2=483$

2024 Turkey EGMO TST, 5
Tags: grid , prime number , combinatorics
Let $p$ be a given prime number. For positive integers $n,k\geq2$ let $S_1, S_2,\dots, S_n$ be unit square sets constructed by choosing exactly one unit square from each of the columns from $p\times k$ chess board. If $|S_i \cap S_j|=1$ for all $1\leq i < j \leq n$ and for any duo of unit squares which are located at different columns there exists $S_i$ such that both of these unit squares are in $S_i$ find all duos of $(n,k)$ in terms of $p$. Note: Here we denote the number of rows by $p$ and the number of columns by $k$.

2010 Contests, 4
Tags: algebra , function , polynomial , number theory , prime number , inequalities
With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

2011 Dutch IMO TST, 4
Tags: number theory , recurrence relation , sequence , prime number
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

2008 Tuymaada Olympiad, 2
Tags: number theory , prime number
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2003 Poland - Second Round, 4
Tags: number theory , combinatorics , combinatorial number theory , prime number , pigeonhole principle
Prove that for any prime number $p > 3$ exist integers $x, y, k$ that meet conditions: $0 < 2k < p$ and $kp + 3 = x^2 + y^2$.

2000 Flanders Math Olympiad, 3
Tags: number theory , prime number
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?