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: 15460

2022 Kosovo Team Selection Test, 2
Tags: number theory
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

2013 Mexico National Olympiad, 1
Tags: number theory , prime number , inequalities
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 $2(a-b)$.

2008 Iran MO (3rd Round), 2
Tags: number theory , arithmetic sequence
Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3\plus{}1\]

2002 Tournament Of Towns, 2
Tags: greatest common divisor , number theory , combinatorics
All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?

1996 Tournament Of Towns, (509) 2
Tags: number theory , divide , divisible
Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

2011 Canadian Students Math Olympiad, 2
Tags: quadratic , number theory
For a fixed positive integer $k$, prove that there exist infinitely many primes $p$ such that there is an integer $w$, where $w^2-1$ is not divisible by $p$, and the order of $w$ in modulus $p$ is the same as the order of $w$ in modulus $p^k$. [i]Author: James Rickards[/i]

2015 Azerbaijan JBMO TST, 2
Tags: number theory
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$

2021 239 Open Mathematical Olympiad, 3
Tags: number theory
Given are two distinct sequences of positive integers $(a_n)$ and $(b_n)$, such that their first two members are coprime and smaller than $1000$, and each of the next members is the sum of the previous two. 8-9 grade Prove that if $a_n$ is divisible by $b_n$, then $n<50$ 10-11 grade Prove that if $a_n^{100}$ is divisible by $b_n$ then $n<5000$

2018-IMOC, C6
Tags: number theory , combinatorics
In a deck of cards, there are $kn$ cards numbered from $1$ to $n$ and there are $k$ cards of each number. Now, divide this deck into $k$ sub-decks with equal sizes. Prove that if $\gcd(k,n)=1$, then one could always pick $n$ cards, one from each sub-deck, such that the sum of those cards is divisible by $n$.

2005 Germany Team Selection Test, 1
Tags: number theory
[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

2015 India IMO Training Camp, 1
Tags: number theory
Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.

2007 Czech-Polish-Slovak Match, 2
Tags: number theory
The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m.$

2017 CMIMC Number Theory, 9
Tags: number theory
Find the smallest prime $p$ for which there exist positive integers $a,b$ such that \[ a^{2} + p^{3} = b^{4}. \]

2020 Taiwan TST Round 2, 1
Tags: number theory
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$.

2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Tags: prime , divide , divisible , number theory
Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

2016 Hanoi Open Mathematics Competitions, 6
Tags: interval , number theory , minimum , algebra
Determine the smallest positive number $a$ such that the number of all integers belonging to $(a, 2016a]$ is $2016$.

1973 Chisinau City MO, 70
Tags: diophantine , diophantine equation , number theory
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

1999 Cono Sur Olympiad, 1
Tags: irreducible , number theory
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.

2000 Swedish Mathematical Competition, 3
Tags: diophantine , diophantine equation , number theory
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

2020-21 IOQM India, 13
Tags: number theory
Find the sum of all positive integers $n$ for which $\mid 2^n + 5^n - 65 \mid$ is a perfect square.

2022 JBMO Shortlist, N6
Tags: sum of squares , digit , shortlist , number theory
Find all positive integers $n$ for which there exists an integer multiple of $2022$ such that the sum of the squares of its digits is equal to $n$.

1967 IMO Longlists, 16
Tags: approximation , irrational number , number theory
Prove the following statement: If $r_1$ and $r_2$ are real numbers whose quotient is irrational, then any real number $x$ can be approximated arbitrarily well by the numbers of the form $\ z_{k_1,k_2} = k_1r_1 + k_2r_2$ integers, i.e. for every number $x$ and every positive real number $p$ two integers $k_1$ and $k_2$ can be found so that $|x - (k_1r_1 + k_2r_2)| < p$ holds.

1978 AMC 12/AHSME, 21
Tags: factorial , number theory
$p$ and $q$ are distinct prime numbers. Prove that the number \[\frac {(pq-1)!} {p^{q-1}q^{p-1}(p-1)!(q-1)!}\] is an integer.

2004 Belarusian National Olympiad, 4
Tags: number theory
For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$.

1963 Czech and Slovak Olympiad III A, 2
Tags: max , relatively prime , number theory
Let an even positive integer $2k$ be given. Find such relatively prime positive integers $x, y$ that maximize the product $xy$.