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

2004 Iran MO (3rd Round), 18
Tags: least common multiple , number theory
Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.

2011 China Northern MO, 3
Tags: number theory , diophantine equation , diophantine
Find all positive integer solutions $(x, y, z)$ of the equation $1 + 2^x \cdot 7^y=z^2$.

2013 Balkan MO Shortlist, N5
Tags: number theory , diophantine equation , diophantine , prime
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

2024 ELMO Shortlist, N2
Tags: number theory
Call a positive integer [i]emphatic[/i] if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all [i]emphatic[/i]. [i]Allen Wang[/i]

2024 IFYM, Sozopol, 7
Tags: number theory
A set \( S \) of two or more positive integers is called [i]almost closed under addition[/i] if the sum of any two distinct elements of \( S \) also belongs to \( S \). Let \( P(x) \) be a polynomial with integer coefficients for which there exists an almost closed under addition set \( S \), such that for any two distinct \( a \) and \( b \) from \( S \), the numbers \( P(a) \) and \( P(b) \) are coprime. Prove that \( P \) is a constant.

2015 British Mathematical Olympiad Round 1, 6
Tags: number theory
A positive integer is called [i]charming[/i] if it is equal to $2$ or is of the form $3^{i}5^{j}$ where $i$ and $j$ are non-negative integers. Prove that every positive integer can be written as a sum of different charming numbers.

2021 USAJMO, 5
Tags: number theory , greatest common divisor
A finite set $S$ of positive integers has the property that, for each $s \in S,$ and each positive integer divisor $d$ of $s$, there exists a unique element $t \in S$ satisfying $\text{gcd}(s, t) = d$. (The elements $s$ and $t$ could be equal.) Given this information, find all possible values for the number of elements of $S$.

2011 Morocco TST, 2
Tags: number theory
For positive integers $m$ and $n$, find the smalles possible value of $|2011^m-45^n|$. [i](Swiss Mathematical Olympiad, Final round, problem 3)[/i]

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 $s^n (F) = F\cup \{a + n|a \in F\}$.

2021 China Team Selection Test, 3
Tags: number theory , additive number theory
Given positive integer $n$. Prove that for any integers $a_1,a_2,\cdots,a_n,$ at least $\lceil \tfrac{n(n-6)}{19} \rceil$ numbers from the set $\{ 1,2, \cdots, \tfrac{n(n-1)}{2} \}$ cannot be represented as $a_i-a_j (1 \le i, j \le n)$.

2014 India IMO Training Camp, 2
Tags: number theory
Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

1949-56 Chisinau City MO, 7
Tags: factorial , number theory , divisible , prime , divide
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2000 Korea Junior Math Olympiad, 5
Tags: number theory , digit
$a$ is a $2000$ digit natural number of the form $$a=2(A)99…99(B)(C)$$ expressed in base $10$. $a$ is not a multiple of $10$, and $2(A)+(B)(C)=99$. $a=2899..9971$ is a possible example of $a$. $b$ is a number you earn when you write the digits of $a$ in a reverse order(Writing the digits of some number in a reverse order means like reordering $1234$ into $4321$). Find every positive integer $a$ that makes $ab$ a square number.

2020 Estonia Team Selection Test, 3
Tags: number theory , polynomial
We say that a set $S$ of integers is [i]rootiful[/i] if, for any positive integer $n$ and any $a_0, a_1, \cdots, a_n \in S$, all integer roots of the polynomial $a_0+a_1x+\cdots+a_nx^n$ are also in $S$. Find all rootiful sets of integers that contain all numbers of the form $2^a - 2^b$ for positive integers $a$ and $b$.

2021/2022 Tournament of Towns, P2
Tags: number theory
Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation?

2007 German National Olympiad, 2
Tags: square , set , product , number theory
Let $A$ be the set of odd integers $\leq 2n-1.$ For a positive integer $m$, let $B=\{a+m\,|\, a\in A \}.$ Determine for which positive integers $n$ there exists a positive integer $m$ such that the product of all elements in $A$ and $B$ is a square.

1986 IMO Longlists, 58
Tags: number theory , prime factorization , divisor
Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2008 ITest, 52
Tags: number theory , relatively prime
A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $c/d$ is the length of a side of the square, find the value of $c+d$.

2016 Switzerland Team Selection Test, Problem 11
Tags: number theory , divisibility
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

2013 HMNT, 6
Tags: number theory
Find the number of positive integer divisors of $12! $ that leave a remainder of $1$ when divided by $3$.

2023 May Olympiad, 2
Tags: number theory
We say that a four-digit number $\overline{abcd}$ is [i]slippery [/i] if the number $a^4+b^3+c^2+d$ is equal to the two-digit number $\overline{cd}$. For example, $2023$ slippery, since $2^4 + 0^3 + 2 ^2 + 3 = 23$. How many slippery numbers are there?

2003 All-Russian Olympiad Regional Round, 8.3
Tags: number theory , combinatorics
Two people take turns writing natural numbers from $1$ to $1000$. On the first move, the first player writes the number $1$ on the board. Then with your next move you can write either the number $2a$ or the number $a+1$ on the board if number $a$ is already written on the board. In this case, it is forbidden to write down numbers that are already written on the board. The one who writes out wins the number $1000$ on the board. Who wins if played correctly?

2019 Kosovo National Mathematical Olympiad, 2
Tags: number theory
Find all positive integers $n$ such that $6^n+1$ it has all the same digits when it is writen in decimal representation.

2009 District Round (Round II), 1
Tags: modular arithmetic , number theory
given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

2000 All-Russian Olympiad, 2
Tags: number theory
Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset.