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

2000 AMC 10, 11
Tags: number theory , prime numbers
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$

2025 Romania National Olympiad, 4
Tags: number theory , prime numbers , algebra , polynomial , finite fields
Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

2012 IMO Shortlist, N3
Tags: number theory , prime numbers , binomial coefficients , IMO Shortlist
Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2017 Dutch IMO TST, 2
Tags: number theory , prime numbers
Let $n \geq 4$ be an integer. Consider a regular $2n-$gon for which to every vertex, an integer is assigned, which we call the value of said vertex. If four distinct vertices of this $2n-$gon form a rectangle, we say that the sum of the values of these vertices is a rectangular sum. Determine for which (not necessarily positive) integers $m$ the integers $m + 1, m + 2, . . . , m + 2n$ can be assigned to the vertices (in some order) in such a way that every rectangular sum is a prime number. (Prime numbers are positive by definition.)

2007 Danube Mathematical Competition, 4
Tags: number theory , prime numbers , number theory proposed
Let $ a,n$ be positive integers such that $ a\ge(n\minus{}1)!$. Prove that there exist $ n$ [i]distinct[/i] prime numbers $ p_1,\ldots,p_n$ so that $ p_i|a\plus{}i$, for all $ i\equal{}\overline{1,\ldots,n}$.

2003 Iran MO (3rd Round), 3
Tags: number theory , prime numbers , number theory unsolved
assume that A is a finite subset of prime numbers, and a is an positive integer. prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.

2017 IFYM, Sozopol, 1
Tags: algebra , number theory , prime numbers
Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

2004 Federal Competition For Advanced Students, P2, 2
Tags: number theory , prime numbers , number theory unsolved
Show that every set $ \{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $ \frac{1}{n}$), whose denominators are exactly the $ k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $ \frac{1}{2004}$ is among this summands? Show that for every set $ \{p_1,p_2,\dots,p_k\}$ containing $ k$ prime numbers ($ k>2$) is the sum smaller than $ \frac{1}{N}$ with $ N=2\cdot 3^{k-2}(k-2)!$

2023 Junior Macedonian Mathematical Olympiad, 2
Tags: number theory , prime numbers
A positive integer is called [i]superprime[/i] if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. [i]Authored by Nikola Velov[/i]

2020 Durer Math Competition Finals, 1
Tags: induction , modular arithmetic , strong induction , number theory , relatively prime , prime numbers , number theory unsolved
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant. [The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]

1977 IMO, 3
Tags: number theory , prime numbers , prime factorization , Dirichlet s Theorem , IMO , IMO 1977
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.)

1990 IMO Shortlist, 27
Tags: number theory , decimal representation , prime numbers , Digits , IMO Shortlist
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2003 Spain Mathematical Olympiad, Problem 1
Tags: number theory , prime numbers
Prove that for any prime ${p}$, different than ${2}$ and ${5}$, there exists such a multiple of ${p}$ whose digits are all nines. For example, if ${p = 13}$, such a multiple is ${999999 = 13 * 76923}$.

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

2012 Czech And Slovak Olympiad IIIA, 1
Tags: number theory , prime numbers , prime
Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime

2013 Mexico National Olympiad, 1
Tags: inequalities , number theory , prime numbers , number theory proposed
All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

LMT Speed Rounds, 2016.6
Tags: number theory , prime numbers
A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. [i]Proposed by Evan Fang

2018 Mediterranean Mathematics OIympiad, 3
Tags: number theory , prime numbers , Mediterranean
An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

2011 Dutch IMO TST, 4
Tags: recurrence relation , Sequence , prime numbers , number theory
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 $p_k = 2p_{k-1} - 1$.

2020 AMC 12/AHSME, 4
Tags: AMC , AMC 10 , prime numbers , 2020 AMC , 2020 AMC 10B , 2020 AMC 12B , AMC 12
The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

1989 IMO, 5
Tags: modular arithmetic , number theory , prime numbers , Sequence , power of number , IMO , IMO 1989
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2019 Macedonia Junior BMO TST, 1
Tags: JMMO , Junior , Macedonia , number theory , prime numbers
Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.

2024 Korea Junior Math Olympiad, 6
Tags: number theory , prime numbers
Find all pairs $(n, p)$ that satisfy the following condition, where $n$ is a positive integer and $p$ is a prime number. [b]Condition)[/b] $2n-1$ is a divisor of $p-1$ and $p$ is a divisor of $4n^2+7$.

1975 Bundeswettbewerb Mathematik, 2
Tags: number theory , prime numbers
Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.

2001 Romania Team Selection Test, 2
Tags: function , pigeonhole principle , number theory , prime numbers , algebra proposed , combinatorics
a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.