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: 71

1978 IMO Longlists, 22
Tags: special factorizations , number theory solved , number theory
Let $x$ and $y$ be two integers not equal to $0$ such that $x+y$ is a divisor of $x^2+y^2$. And let $\frac{x^2+y^2}{x+y}$ be a divisor of $1978$. Prove that $x = y$. [i]German IMO Selection Test 1979, problem 2[/i]

2004 France Team Selection Test, 1
Tags: quadratics , modular arithmetic , number theory , number theory solved
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$. Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?

1998 India National Olympiad, 2
Tags: quadratics , algebra , number theory solved , number theory
Let $a$ and $b$ be two positive rational numbers such that $\sqrt[3] {a} + \sqrt[3]{b}$ is also a rational number. Prove that $\sqrt[3]{a}$ and $\sqrt[3] {b}$ themselves are rational numbers.

2003 Romania Team Selection Test, 10
Tags: modular arithmetic , pigeonhole principle , inequalities , number theory , relatively prime , number theory solved
Let $\mathcal{P}$ be the set of all primes, and let $M$ be a subset of $\mathcal{P}$, having at least three elements, and such that for any proper subset $A$ of $M$ all of the prime factors of the number $ -1+\prod_{p\in A}p$ are found in $M$. Prove that $M= \mathcal{P}$. [i]Valentin Vornicu[/i]

2022 All-Russian Olympiad, 1
Tags: number theory , All russia mo , All Russian Olympiad , Russia , Russian , number theory solved , Divisors
We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

2021 JBMO TST - Turkey, 2
Tags: number theory solved , Turkey
For which positive integers $n$, one can find a non-integer rational number $x$ such that $$x^n+(x+1)^n$$ is an integer?

2004 India IMO Training Camp, 4
Tags: function , arithmetic sequence , number theory solved , number theory , BritishMathematicalOlympiad
Let $f$ be a bijection of the set of all natural numbers on to itself. Prove that there exists positive integers $a < a+d < a+ 2d$ such that $f(a) < f(a+d) <f(a+2d)$

2004 Korea - Final Round, 3
Tags: number theory solved , number theory
For prime number $p$, let $f_p(x)=x^{p-1} +x^{p-2} + \cdots + x + 1$. (1) When $p$ divides $m$, prove that there exists a prime number that is coprime with $m(m-1)$ and divides $f_p(m)$. (2) Prove that there are infinitely many positive integers $n$ such that $pn+1$ is prime number.

2006 CHKMO, 4
Tags: search , number theory solved , number theory
Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

2004 Romania Team Selection Test, 6
Tags: invariant , symmetry , quadratics , algebra , polynomial , Vieta , number theory solved
Let $a,b$ be two positive integers, such that $ab\neq 1$. Find all the integer values that $f(a,b)$ can take, where \[ f(a,b) = \frac { a^2+ab+b^2} { ab- 1} . \]

1988 IMO Longlists, 86
Tags: number theory solved , number theory
Let $a,b,c$ be integers different from zero. It is known that the equation $a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 0$ has a solution $(x,y,z)$ in integer numbers different from the solutions $x = y = z = 0.$ Prove that the equation \[ a \cdot x^2 + b \cdot y^2 + c \cdot z^2 = 1 \] has a solution in rational numbers.

2000 South africa National Olympiad, 1
Tags: number theory solved , number theory
A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.

1998 South africa National Olympiad, 5
Tags: number theory , greatest common divisor , number theory solved
Prove that \[ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} \] is a prime if $n$ is a power of a prime, and 1 otherwise.

2004 India IMO Training Camp, 2
Tags: floor function , number theory solved , number theory
Find all primes $p \geq 3$ with the following property: for any prime $q<p$, the number \[ p - \Big\lfloor \frac{p}{q} \Big\rfloor q \] is squarefree (i.e. is not divisible by the square of a prime).

1996 India National Olympiad, 1
Tags: modular arithmetic , number theory solved , number theory
a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$; b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.

2005 India IMO Training Camp, 2
Tags: geometry , 3D geometry , number theory solved , number theory
Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

2004 France Team Selection Test, 3
Tags: induction , number theory , prime numbers , number theory solved
Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2005 Hong kong National Olympiad, 3
Tags: search , number theory solved , number theory
Show that there exist infinitely many square-free positive integers $n$ that divide $2005^n-1$.

2005 Postal Coaching, 4
Tags: number theory , greatest common divisor , algorithm , Euclidean algorithm , number theory solved
Let $m,n$ be natural numbers and let $d = gcd(m,n)$. Let $x = 2^{m} -1$ and $y= 2^n +1$ (a) If $\frac{m}{d}$ is odd, prove that $gcd(x,y) = 1$ (b) If $\frac{m}{d}$ is even, Find $gcd(x,y)$

2005 India IMO Training Camp, 2
Tags: geometry , 3D geometry , number theory solved , number theory
Prove that one can find a $n_{0} \in \mathbb{N}$ such that $\forall m \geq n_{0}$, there exist three positive integers $a$, $b$ , $c$ such that (i) $m^3 < a < b < c < (m+1)^3$; (ii) $abc$ is the cube of an integer.

2016 Israel Team Selection Test, 4
Tags: greatest common divisor , number theory , number theory solved
Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.