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: 1766

2010 JBMO Shortlist, 1
Tags: induction , inequalities , number theory , prime factorization , number theory proposed
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

1997 Vietnam National Olympiad, 2
Tags: induction , number theory proposed , number theory
Prove that for evey positive integer n, there exits a positive integer k such that $ 2^n | 19^k \minus{} 97$

1976 Canada National Olympiad, 5
Tags: search , number theory proposed , number theory
Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.

2012 India IMO Training Camp, 2
Tags: number theory , greatest common divisor , number theory proposed
Show that there exist infinitely many pairs $(a, b)$ of positive integers with the property that $a+b$ divides $ab+1$, $a-b$ divides $ab-1$, $b>1$ and $a>b\sqrt{3}-1$

2014 Contests, 2
Tags: floor function , limit , number theory proposed , number theory
$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

2012 Greece Team Selection Test, 1
Tags: number theory proposed , number theory
Find all triples $(p,m,n)$ satisfying the equation $p^m-n^3=8$ where $p$ is a prime number and $m,n$ are nonnegative integers.

2008 China Team Selection Test, 2
Tags: modular arithmetic , number theory , relatively prime , number theory proposed
Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

2014 Korea National Olympiad, 1
Tags: number theory , greatest common divisor , number theory proposed
For $x, y$ positive integers, $x^2-4y+1$ is a multiple of $(x-2y)(1-2y)$. Prove that $|x-2y|$ is a square number.

2017 German National Olympiad, 6
Tags: number theory , number theory proposed , Germany , Diophantine equation , Diophantine Equations
Prove that there exist infinitely many positive integers $m$ such that there exist $m$ consecutive perfect squares with sum $m^3$. Specify one solution with $m>1$.

2011 Canadian Mathematical Olympiad Qualification Repechage, 8
Tags: induction , greatest common divisor , number theory proposed , number theory
Determine all pairs $(n,m)$ of positive integers for which there exists an infinite sequence $\{x_k\}$ of $0$'s and $1$'s with the properties that if $x_i=0$ then $x_{i+m}=1$ and if $x_i = 1$ then $x_{i+n} = 0.$

2013 Bulgaria National Olympiad, 6
Tags: number theory , Diophantine equation , number theory proposed
Given $m\in\mathbb{N}$ and a prime number $p$, $p>m$, let \[M=\{n\in\mathbb{N}\mid m^2+n^2+p^2-2mn-2mp-2np \,\,\, \text{is a perfect square} \} \] Prove that $|M|$ does not depend on $p$. [i]Proposed by Aleksandar Ivanov[/i]

1997 Baltic Way, 9
Tags: geometry , 3D geometry , sphere , number theory proposed , number theory
The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?

1999 Greece National Olympiad, 2
Tags: geometry , number theory , number theory proposed
A right triangle has integer side lengths, and the sum of its area and the length of one of its legs equals $75$. Find the side lengths of the triangle.

2014 Postal Coaching, 3
Tags: modular arithmetic , number theory proposed , number theory
Find all ordered triplets of positive integers $(a,\ b,\ c)$ such that $2^a+3^b+1=6^c$.

2016 Bundeswettbewerb Mathematik, 1
Tags: number theory , number theory proposed , consecutive , Sum , Summation
There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

2001 ITAMO, 4
Tags: least common multiple , number theory proposed , number theory
A positive integer is called [i]monotone[/i] if has at least two digits and all its digits are nonzero and appear in a strictly increasing or strictly decreasing order. (a) Compute the sum of all monotone five-digit numbers. (b) Find the number of final zeros in the least common multiple of all monotone numbers (with any number of digits).

2002 India IMO Training Camp, 8
Tags: number theory proposed , number theory
Let $\sigma(n)=\sum_{d|n} d$, the sum of positive divisors of an integer $n>0$. [list] [b](a)[/b] Show that $\sigma(mn)=\sigma(m)\sigma(n)$ for positive integers $m$ and $n$ with $gcd(m,n)=1$ [b](b)[/b] Find all positive integers $n$ such that $\sigma(n)$ is a power of $2$.[/list]

2011 Federal Competition For Advanced Students, Part 1, 1
Tags: quadratics , modular arithmetic , calculus , integration , number theory , prime factorization , number theory proposed
Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

2014 Cono Sur Olympiad, 4
Tags: number theory proposed , number theory
Show that the number $n^{2} - 2^{2014}\times 2014n + 4^{2013} (2014^{2}-1)$ is not prime, where $n$ is a positive integer.

2008 Portugal MO, 3
Tags: number theory proposed , number theory
Let $d$ be a natural number. Given two natural numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if the $d$ numbers obtained substituting each one of the digits of $M$ by the digit of $N$ which is on the same position are all multiples of $7$. Find all the values of $d$ for which the following condition is valid: For any two numbers $M$ and $N$ with $d$ digits, $M$ is a friend of $N$ if and only if $N$ is a friend of $M$.

2024 All-Russian Olympiad, 1
Tags: number theory , number theory proposed
Petya and Vasya only know positive integers not exceeding $10^9-4000$. Petya considers numbers as good which are representable in the form $abc+ab+ac+bc$, where $a,b$ and $c$ are natural numbers not less than $100$. Vasya considers numbers as good which are representable in the form $xyz-x-y-z$, where $x,y$ and $z$ are natural numbers strictly bigger than $100$. For which of them are there more good numbers? [i]Proposed by I. Bogdanov[/i]

1994 Iran MO (2nd round), 1
Tags: function , pigeonhole principle , modular arithmetic , number theory proposed , number theory
Let $\overline{a_1a_2a_3\ldots a_n}$ be the representation of a $n-$digits number in base $10.$ Prove that there exists a one-to-one function like $f : \{0, 1, 2, 3, \ldots, 9\} \to \{0, 1, 2, 3, \ldots, 9\}$ such that $f(a_1) \neq 0$ and the number $\overline{ f(a_1)f(a_2)f(a_3) \ldots f(a_n) }$ is divisible by $3.$

1999 Baltic Way, 17
Tags: number theory proposed , number theory
Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?

2024 Baltic Way, 18
Tags: number theory , number theory proposed , Sequence , Divisors , prime numbers , Bound
An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

2003 Bulgaria Team Selection Test, 4
Tags: number theory proposed , number theory
Is it true that for any permulation $a_1,a_2.....,a_{2002}$ of $1,2....,2002$ there are positive integers $m,n$ of the same parity such that $0<m<n<2003$ and $a_m+a_n=2a_{\frac {m+n}{2}}$