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.

AND:
OR:
NO:

Found problems: 15460

1991 Swedish Mathematical Competition, 1
Tags: diophantine , diophantine equation , number theory
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

2024 Regional Olympiad of Mexico West, 5
Tags: recurrence relation , number theory
Consider a sequence of positive integers $a_1,a_2,a_3,...$ such that $a_1>1$ and $$a_{n+1}=\frac{a_n}{p}+p,$$ where $p$ is the greatest prime factor of $a_n$. Prove that for any choice of $a_1$, the sequence $a_1,a_2,a_3,...$ has an infinite terms that are equal between them.

2016 Korea Winter Program Practice Test, 1
Tags: diophantine equation , number theory
Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

2013 Danube Mathematical Competition, 3
Tags: exponential , number theory , diophantine , diophantine equation , exponential equation
Determine the natural numbers $m,n$ such as $85^m-n^4=4$

2004 Bosnia and Herzegovina Team Selection Test, 2
Tags: geometry , area , number theory
Determine whether does exists a triangle with area $2004$ with his sides positive integers.

1965 Bulgaria National Olympiad, Problem 1
Tags: number theory
The numbers $2,3,7$ have the property that the product of any two of them increased by $1$ is divisible by the third number. Prove that this triple of integer numbers greater than $1$ is the only triple with the given property.

2007 Indonesia TST, 3
Tags: inequalities , diophantine equation , floor function , number theory
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

MathLinks Contest 4th, 1.2
Tags: number theory
Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.

1964 IMO, 1
Tags: number theory
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.

2013 Brazil Team Selection Test, 4
Tags: algebra , rational roots , number theory , real analysis , polynomial
Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

2014 AIME Problems, 12
Tags: function , algebra , domain , probability , number theory
Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

2004 Brazil Team Selection Test, Problem 2
Tags: algebra , number theory
Let $(x+1)^p(x-3)^q=x^n+a_1x^{n-1}+\ldots+a_{n-1}x+a_n$, where $p,q$ are positive integers. (a) Prove that if $a_1=a_2$, then $3n$ is a perfect square. (b) Prove that there exists infinitely many pairs $(p,q)$ for which $a_1=a_2$.

2022 Serbia Team Selection Test, P3
Tags: double counting , combinatorics , number theory
Let $n$ be an odd positive integer. Given are $n$ balls - black and white, placed on a circle. For a integer $1\leq k \leq n-1$, call $f(k)$ the number of balls, such that after shifting them with $k$ positions clockwise, their color doesn't change. a) Prove that for all $n$, there is a $k$ with $f(k) \geq \frac{n-1}{2}$. b) Prove that there are infinitely many $n$ (and corresponding colorings for them) such that $f(k)\leq \frac{n-1}{2}$ for all $k$.

2015 USAMO, 1
Tags: diophantine equation , number theory , algebra
Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

2017 QEDMO 15th, 5
Tags: number theory
Let $F$ be a finite subset of the integer numbers. We define a new subset $s(F)$ in that $a\in Z$ lies in $s (F)$ if and only if exactly one of the numbers $a$ and $a -1$ in $F$. In the same way one gets from $s (F)$ the set $s^2(F) = s (s (F))$ and by $n$-fold application of $s$ then iteratively further subsets $s^n (F)$. Prove there are infinitely many natural numbers $n$ for which $s^n (F) = F\cup \{a + n|a \in F\}$.

2011 Iran MO (3rd Round), 1
Tags: number theory , algorithm
Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true? There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$. [i]proposed by Mahyar Sefidgaran[/i]

2000 Mexico National Olympiad, 4
Tags: integer sequence , maximum , number theory , prime
Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?

2019 Azerbaijan Junior NMO, 3
Tags: difference of squares , number theory
A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

2014 Denmark MO - Mohr Contest, 1
Tags: algebra , number theory
Georg chooses three distinct digits among $1, 2, . . . , 9$ and writes them down on three cards. When the cards are laid down next to each other, a three-digit number is formed. Georg tells his mother that the sum of the largest and the second-largest number that can be formed in this manner is $1732$. Can she figure out which three digits Georg has chosen?

1973 Swedish Mathematical Competition, 4
Tags: diophantine , number theory , diophantine equation
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that \[ \frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p} \]

1995 Tournament Of Towns, (444) 4
Tags: perfect square , number theory
Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square. (VA Senderov)

2018 Iran MO (3rd Round), 4
Tags: number theory
Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)

1975 Chisinau City MO, 116
Tags: trigonometry , radical , number theory
The sides of a triangle are equal to $\sqrt2, \sqrt3, \sqrt4$ and its angles are $\alpha, \beta, \gamma$, respectively. Prove that the equation $x\sin \alpha + y\sin \beta + z\sin \gamma = 0$ has exactly one solution in integers $x, y, z$.

2005 Tournament of Towns, 2
Tags: number theory
Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$. [i](4 points)[/i]

2019 Romania National Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$