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

1979 IMO Longlists, 55
Tags: greatest common divisor , search , relatively prime , number theory
Let $a,b$ be coprime integers. Show that the equation $ax^2 + by^2 =z^3$ has an infinite set of solutions $(x,y,z)$ with $\{x,y,z\}\in\mathbb{Z}$ and each pair of $x,y$ mutually coprime.

1996 Tournament Of Towns, (484) 2
Tags: number theory , perfect square
Does there exist an integer n such that all three numbers (a) $n - 96$, $n$ and $n + 96$ (b) $n - 1996$, $n$ and $n + 1996$ are positive prime numbers? (V Senderov)

2002 Junior Balkan Team Selection Tests - Romania, 1
Tags: number theory , diophantine equation , diophantine
Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

2015 Romanian Master of Mathematics, 1
Tags: number theory , relatively prime
Does there exist an infinite sequence of positive integers $a_1, a_2, a_3, . . .$ such that $a_m$ and $a_n$ are coprime if and only if $|m - n| = 1$?

2017 May Olympiad, 1
Tags: number theory , digit , odd
To each three-digit number, Matías added the number obtained by inverting its digits. For example, he added $729$ to the number $927$. Calculate in how many cases the result of the sum of Matías is a number with all its digits odd.

2016 Hanoi Open Mathematics Competitions, 8
Tags: number theory , diophantine equation , diophantine
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$

1993 IMO Shortlist, 2
Tags: modular arithmetic , number theory , divisibility , prime number , composite number
A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

2019 HMNT, 2
Tags: number theory
Meghana writes two (not necessarily distinct) primes $q$ and $r$ in base $10$ next to each other on a blackboard, resulting in the concatenation of $q$ and $r$ (for example, if $q = 13$ and $r = 5$, the number on the blackboard is now $135$). She notices that three more than the resulting number is the square of a prime $p$. Find all possible values of $p$.

2020 Dutch IMO TST, 1
Tags: number theory , number of divisors
For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

2016 Brazil Undergrad MO, 3
Tags: number theory , recurrence relation
Let it \(k \geq 1\) be an integer. Define the sequence \((a_n)_{n \geq 1}\) by \(a_0=0,a_1=1\) and \[ a_{n+2} = ka_{n+1}+a_n \] for \(n \geq 0\). Let it \(p\) an odd prime number. Denote \(m(p)\) as the smallest positive integer \(m\) such that \(p | a_m\). Denote \(T(p)\) as the smallest positive integer \(T\) such that for every natural \(j\) we gave \(p | (a_{T+j}-a_j)\). [list='i'] [*] Show that \(T(p) \leq (p-1) \cdot m(p)\). [*] Show that if \(T(p) = (p-1) \cdot m(p)\) then \[ \prod_{1 \leq j \leq T(p)-1}^{j \not \equiv 0 \pmod{m(p)}}{a_j} \equiv (-1)^{m(p)-1} \pmod{p} \] [/list]

2006 Tuymaada Olympiad, 1
Tags: number theory
Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ? [i]Proposed by F. Petrov, K. Sukhov[/i]

2014 Canada National Olympiad, 3
Tags: number theory
Let $p$ be a fixed odd prime. A $p$-tuple $(a_1,a_2,a_3,\ldots,a_p)$ of integers is said to be [i]good[/i] if [list] [*] [b](i)[/b] $0\le a_i\le p-1$ for all $i$, and [*] [b](ii)[/b] $a_1+a_2+a_3+\cdots+a_p$ is not divisible by $p$, and [*] [b](iii)[/b] $a_1a_2+a_2a_3+a_3a_4+\cdots+a_pa_1$ is divisible by $p$.[/list] Determine the number of good $p$-tuples.

1989 IMO Longlists, 88
Tags: number theory
Prove that the sequence $ (a_n)_{n \geq 0,}, a_n \equal{} [n \cdot \sqrt{2}],$ contains an infinite number of perfect squares.

2018 Purple Comet Problems, 22
Tags: number theory
Positive integers $a$ and $b$ satisfy $a^3 + 32b + 2c = 2018$ and $b^3 + 32a + 2c = 1115$. Find $a^2 + b^2 + c^2$.

2016 Czech-Polish-Slovak Junior Match, 6
Tags: diophantine equation , system of equations , diophantine , number theory
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that $a + b + c = 3k + 1$, $ab + bc + ca = 3k^2 + 2k$. Slovakia

2022 VN Math Olympiad For High School Students, Problem 5
Tags: algebra , number theory
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $p$ is an odd prime such that $p \equiv \pm 1(\bmod 5)$. Prove that: a) ${5^{\frac{{p - 1}}{2}}} \equiv 1(\bmod p).$ b) ${F_{p - 1}} \equiv 0(\bmod p).$ c) $k(p)|p-1.$

2001 Brazil Team Selection Test, Problem 4
Tags: algebra , number theory
Prove that for all integers $n\ge3$ there exists a set $A_n=\{a_1,a_2,\ldots,a_n\}$ of $n$ distinct natural numbers such that, for each $i=1,2,\ldots,n$, $$\prod_{\small{\begin{matrix}1\le k\le n\\k\ne i\end{matrix}}}a_k\equiv1\pmod{a_i}.$$

2023 USEMO, 1
Tags: number theory
A positive integer $n$ is called [i]beautiful[/i] if, for every integer $4 \le b \le 10000$, the base-$b$ representation of $n$ contains the consecutive digits $2$, $0$, $2$, $3$ (in this order, from left to right). Determine whether the set of all beautiful integers is finite. [i]Oleg Kryzhanovsky[/i]

2020 Harvest Math Invitational Team Round Problems, HMI Team #6
Tags: combi , number theory
6. A triple of integers $(a,b,c)$ is said to be $\gamma$[i]-special[/i] if $a\le \gamma(b+c)$, $b\le \gamma(c+a)$ and $c\le\gamma(a+b)$. For each integer triple $(a,b,c)$ such that $1\le a,b,c \le 20$, Kodvick writes down the smallest value of $\gamma$ such that $(a,b,c)$ is $\gamma$-special. How many distinct values does he write down? [i]Proposed by winnertakeover[/i]

2020 Saint Petersburg Mathematical Olympiad, 4.
Tags: number theory
The sum $\frac{2}{3\cdot 6} +\frac{2\cdot 5}{3\cdot 6\cdot 9} +\ldots +\frac{2\cdot5\cdot \ldots \cdot 2015}{3\cdot 6\cdot 9\cdot \ldots \cdot 2019}$ is written as a decimal number. Find the first digit after the decimal point.

2018 Stars of Mathematics, 1
Tags: number theory
Two natural numbers have the property that the product of their positive divisors are equal. Does this imply that they are equal? [i]Proposed by Belarus for the 1999th IMO[/i]

2016 Saint Petersburg Mathematical Olympiad, 7
Tags: integer sequence , sequence , number theory , number theory with sequences
A sequence of $N$ consecutive positive integers is called [i]good [/i] if it is possible to choose two of these numbers so that their product is divisible by the sum of the other $N-2$ numbers. For which $N$ do there exist infinitely many [i]good [/i] sequences?

1998 Estonia National Olympiad, 4
Tags: floor function , number theory , algebra , irrational
A real number $a$ satisfies the equality $\frac{1}{a} = a - [a]$. Prove that $a$ is irrational.

2003 China Team Selection Test, 2
Tags: modular arithmetic , quadratic , number theory
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.

2014 European Mathematical Cup, 4
Tags: function , induction , number theory
Find all functions $f$ from positive integers to themselves such that: 1)$f(mn)=f(m)f(n)$ for all positive integers $m, n$ 2)$\{1, 2, ..., n\}=\{f(1), f(2), ... f(n)\}$ is true for infinitely many positive integers $n$.