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

2014 Indonesia MO Shortlist, N2
Tags: number theory , composite number , composite , diophantine equation
Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.

2004 India IMO Training Camp, 2
Tags: floor function , 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).

2022 Auckland Mathematical Olympiad, 2
Tags: algebra , number theory
The number $12$ is written on the whiteboard. Each minute, the number on the board is either multiplied or divided by one of the numbers $2$ or $3$ (a division is possible only if the result is an integer) . Prove that the number that will be written on the board in exactly one hour will not be equal to $54$.

2019 Junior Balkan Team Selection Tests - Moldova, 1
Tags: number theory
Let $n$ be a positive integer. From the set $A=\{1,2,3,...,n\}$ an element is eliminated. What's the smallest possible cardinal of $A$ and the eliminated element, since the arithmetic mean of left elements in $A$ is $\frac{439}{13}$.

2022 Thailand Mathematical Olympiad, 7
Tags: number theory , sequence
Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by $$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$ Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$.

2011 Hanoi Open Mathematics Competitions, 5
Tags: number theory
Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?

2022 Kurschak Competition, 2
Tags: number theory , prime number , square
Let $p$ and $q$ be prime numbers of the form $4k+3$. Suppose that there exist integers $x$ and $y$ such that $x^2-pqy^2=1$. Prove that there exist positive integers $a$ and $b$ such that $|pa^2-qb^2|=1$.

2013 Silk Road, 1
Tags: greatest common divisor , number theory
Determine all pairs of positive integers $m, n,$ satisfying the equality $(2^{m}+1;2^n+1)=2^{(m;n)}+1$ , where $(a;b)$ is the greatest common divisor

1980 IMO Shortlist, 19
Tags: number theory , sequence , recurrence relation
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

2019 Jozsef Wildt International Math Competition, W. 19
Tags: number theory , fibonacci sequence , lucas sequence
Let $\{F_n\}_{n\in\mathbb{Z}}$ and $\{L_n\}_{n\in\mathbb{Z}}$ denote the Fibonacci and Lucas numbers, respectively, given by $$F_{n+1} = F_n + F_{n-1}\ \text{and}\ L_{n+1} = L_n + L_{n-1}\ \text{for all}\ n \geq 1$$with $F_0 = 0$, $F_1 = 1$, $L_0 = 2$, and $L_1 = 1$. Prove that for integers $n \geq 1$ and $j \geq 0$ [list=1] [*]$\sum \limits_{k=1}^n F_{k\pm j}L_{k\mp j}=F_{2n+1}-1+\begin{cases} 0, & \text{if}\ n\ \text{is even}\\ \left(-1\right)^{\pm j}F_{\pm 2j}, & \text{if}\ n\ \text{is odd} \end{cases}$ [*] $\sum \limits_{k=1}^nF_{k+j}F_{k-j}L_{k+j}L_{k-j}=\frac{F_{4n+2}-1-nL_{4j}}{5}$ [/list]

2012 Indonesia TST, 4
Tags: number theory
Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

2012 ELMO Shortlist, 3
Tags: number theory , additive number theory , extremal combinatorics , perfect power
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. [i]Alex Zhu.[/i]

2001 Portugal MO, 3
Tags: digit , number theory
How many consecutive zeros are there at the end of the number $2001! = 2001 \times 2000 \times ... \times 3 \times 2 \times 1$ ?

2021 Regional Olympiad of Mexico Center Zone, 6
Tags: algebra , number theory , functional equation , sequence
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: [list] [*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ [*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ [/list] Prove that $a_n=n$ for all positive integers $n$. [i]Proposed by José Alejandro Reyes González[/i]

2014 Cuba MO, 4
Tags: number theory , integer
Find all positive integers $a, b$ such that the numbers $\frac{a^2b + a}{a^2 + b}$ and $\frac{ab^2 + b}{b^2 - a}$ are integers.

2009 Hanoi Open Mathematics Competitions, 2
Tags: number theory
Show that there is a natural number $n$ such that the number $a = n!$ ends exactly in $2009$ zeros.

MathLinks Contest 7th, 1.2
Tags: algebra , polynomial , number theory , greatest common divisor , quadratic , calculus , integration
Let $ a,b,c,d$ be four distinct positive integers in arithmetic progression. Prove that $ abcd$ is not a perfect square.

2019 Benelux, 4
Tags: number theory
An integer $m>1$ is [i]rich[/i] if for any positive integer $n$, there exist positive integers $x,y,z$ such that $n=mx^2-y^2-z^2$. An integer $m>1$ is [i]poor[/i] if it is not rich. [list=a] [*]Find a poor integer.[/*] [*]Find a rich integer.[/*] [/list]

2021 Macedonian Balkan MO TST, Problem 2
Tags: number theory , sequence
Define a sequence: $x_0=1$ and for all $n\ge 0$, $x_{2n+1}=x_{n}$ and $x_{2n+2}=x_{n}+x_{n+1}$. Prove that for any relatively prime positive integers $a$ and $b$, there is a non-negative integer $n$ such that $a=x_n$ and $b=x_{n+1}$.

2002 Mediterranean Mathematics Olympiad, 1
Tags: search , number theory
Find all natural numbers $ x,y$ such that $ y| (x^{2}+1)$ and $ x^{2}| (y^{3}+1)$.

IV Soros Olympiad 1997 - 98 (Russia), 11.2
Tags: number theory , algebra , inequalities
Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

2008 Flanders Math Olympiad, 1
Tags: number theory , digit
Determine all natural numbers $n$ of $4$ digits whose quadruple minus the number formed by the digits of $n$ in reverse order equals $30$.

1996 Poland - Second Round, 5
Tags: number theory , diophantine , diophantine equation
Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

2009 Balkan MO Shortlist, N1
Tags: number theory
Solve the given equation in integers \begin{align*} y^3=8x^6+2x^3y-y^2 \end{align*}

2002 AIME Problems, 8
Tags: number theory , relatively prime
Find the smallest integer $k$ for which the conditions $(1)$ $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers $(2)$ $a_n=a_{n-1}+a_{n-2}$ for all $n>2$ $(3)$ $a_9=k$ are satisfied by more than one sequence.