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

2019 Regional Olympiad of Mexico Southeast, 6
Tags: number theory , prime number
Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.

2022 SG Originals, Q5
Tags: number theory , polynomial , modulo , algebra , prime number
Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

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?

2009 All-Russian Olympiad, 6
Tags: number theory , prime number , combinatorics
Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?

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 $a,b$.

2021/2022 Tournament of Towns, P7
Tags: number theory , prime number , geometry
Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.

1977 IMO, 3
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.)

1996 Bulgaria National Olympiad, 1
Tags: number theory , prime number
Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.

2019 Macedonia Junior BMO TST, 5
Tags: number theory , prime number
Let $p_{1}$, $p_{2}$, ..., $p_{k}$ be different prime numbers. Determine the number of positive integers of the form $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$, $\alpha_{i}$ $\in$ $\mathbb{N}$ for which $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$.

2013 Puerto Rico Team Selection Test, 3
Tags: number theory , prime number
Find all pairs of natural numbers n and prime numbers p such that $\sqrt{n+\frac{p}{n}}$ is a natural number.

2015 Korea National Olympiad, 4
Tags: inequalities , number theory , relatively prime , prime number , coprime integers
For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

2004 Iran MO (3rd Round), 21
Tags: greatest common divisor , modular arithmetic , prime number , number theory
$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$ \[ p\mid a_1^k \plus{} \dots \plus{} a_n^k.\]

2023 Romanian Master of Mathematics, 1
Tags: number theory , prime number , diophantine equation
Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

2010 China Team Selection Test, 2
Tags: number theory , prime number , factorial
Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

2024 Bangladesh Mathematical Olympiad, P1
Tags: number theory , diophantine equation , modular arithmetic , prime number
Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

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?

2014 Poland - Second Round, 6.
Tags: number theory , prime number
Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.

2018 Chile National Olympiad, 1
Tags: number theory , prime number , prime
Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

2007 May Olympiad, 4
Tags: combinatorics , number theory , prime number
Alex and Bruno play the following game: each one, in your turn, the player writes, exactly one digit, in the right of the last number written. The game finishes if we have a number with $6$ digits( distincts ) and Alex starts the game. Bruno wins if the number with $6$ digits is a prime number, otherwise Alex wins. Which player has the winning strategy?

2022 Kosovo National Mathematical Olympiad, 3
Tags: number theory , prime number
Let $a,b$ and $c$ be positive integers such that $a!+b+c,b!+c+a$ and $c!+a+b$ are prime numbers. Show that $\frac{a+b+c+1}{2}$ is also a prime number.

2000 AMC 12/AHSME, 6
Tags: number theory , prime number
Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

2022 IMC, 6
Tags: prime number , modular arithmetic , number theory
Let $p \geq 3$ be a prime number. Prove that there is a permutation $(x_1,\ldots, x_{p-1})$ of $(1,2,\ldots,p-1)$ such that $x_1x_2 + x_2x_3 + \cdots + x_{p-2}x_{p-1} \equiv 2 \pmod p$.

2017 Bosnia And Herzegovina - Regional Olympiad, 3
Tags: number theory , prime number
Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

2016 Bundeswettbewerb Mathematik, 1
Tags: prime number , number theory , decimal representation
A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

2005 Slovenia National Olympiad, Problem 2
Tags: number theory , prime number
For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?