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

2008 IMAR Test, 2
Tags: analytic geometry , Putnam , number theory , prime numbers , number theory proposed
A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2008 Pan African, 3
Tags: number theory proposed , number theory
Prove that for all positive integers $n$, there exists a positive integer $m$ which is a multiple of $n$ and the sum of the digits of $m$ is equal to $n$.

2004 Iran MO (3rd Round), 28
Tags: number theory proposed , number theory
Find all prime numbers $p$ such that $ p = m^2 + n^2$ and $p\mid m^3+n^3-4$.

2014 Contests, 2
Tags: floor function , number theory proposed , number theory
Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]

2001 JBMO ShortLists, 3
Tags: modular arithmetic , number theory , number theory proposed
Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$.

2021 German National Olympiad, 6
Tags: number theory , number theory proposed , Arithmetic Progression , Perfect Squares , Perfect Square
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.

2020 Baltic Way, 19
Tags: number theory , number theory proposed , divisor function
Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.

2009 Baltic Way, 9
Tags: modular arithmetic , inequalities , number theory proposed , number theory
Determine all positive integers $n$ for which $2^{n+1}-n^2$ is a prime number.

2004 Kazakhstan National Olympiad, 3
Tags: modular arithmetic , number theory proposed , number theory
Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

2005 Germany Team Selection Test, 3
Tags: number theory proposed , number theory
A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$. Calculate the sum of the first $ 2005$ nice positive integers.

2015 Miklos Schweitzer, 4
Tags: number theory , number theory proposed , limits , limit , algebra and number theory , prime numbers
Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

2014 All-Russian Olympiad, 1
Tags: number theory proposed , number theory
Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?

2005 All-Russian Olympiad Regional Round, 11.7
Tags: inequalities , number theory proposed , number theory
11.7 Let $N$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 7, and $M$ be a number of perfect squares from $\{1,2,...,10^{20}\}$, which 17-th digit from the end is 8. Compare $M$ and $N$. ([i]A. Golovanov[/i])

2024 Dutch IMO TST, 1
Tags: number theory , number theory proposed , Divisors , coprime
For a positive integer $n$, let $\alpha(n)$ be the arithmetic mean of the divisors of $n$, and let $\beta(n)$ be the arithmetic mean of the numbers $k \le n$ with $\text{gcd}(k,n)=1$. Determine all positive integers $n$ with $\alpha(n)=\beta(n)$.

2014 Postal Coaching, 5
Tags: number theory proposed , number theory , Operation
Fix positive integers $n$ and $k\ge 2$. A list of $n$ integers is written in a row on a blackboard. You can choose a contiguous block of integers, and I will either add $1$ to all of them or subtract $1$ from all of them. You can repeat this step as often as you like, possibly adapting your selections based on what I do. Prove that after a finite number of steps, you can reach a state where at least $n-k+2$ of the numbers on the blackboard are all simultaneously divisible by $k$.

1986 IMO Longlists, 77
Tags: number theory proposed , number theory
Find all integers $x,y,z$ such that \[x^3+y^3+z^3=x+y+z=8\]

2003 Iran MO (2nd round), 1
Tags: number theory proposed , number theory
We call the positive integer $n$ a $3-$[i]stratum[/i] number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others. $a)$ Find a $3-$stratum number. $b)$ Prove that there are infinitely many $3-$stratum numbers.

2011 Balkan MO, 3
Tags: ratio , floor function , modular arithmetic , number theory proposed , number theory
Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

2007 Indonesia TST, 3
Tags: function , algebra , polynomial , induction , number theory proposed , number theory
Find all pairs of function $ f: \mathbb{N} \rightarrow \mathbb{N}$ and polynomial with integer coefficients $ p$ such that: (i) $ p(mn) \equal{} p(m)p(n)$ for all positive integers $ m,n > 1$ with $ \gcd(m,n) \equal{} 1$, and (ii) $ \sum_{d|n}f(d) \equal{} p(n)$ for all positive integers $ n$.

2010 Contests, 4
Tags: inequalities , number theory proposed , number theory , easy problem
Find all positive integers $n$ which satisfy the following tow conditions: (a) $n$ has at least four different positive divisors; (b) for any divisors $a$ and $b$ of $n$ satisfying $1<a<b<n$, the number $b-a$ divides $n$. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 4)[/i]

2007 Romania National Olympiad, 3
Tags: number theory proposed , number theory
For which integers $n\geq 2$, the number $(n-1)^{n^{n+1}}+(n+1)^{n^{n-1}}$ is divisible by $n^{n}$ ?

2009 India IMO Training Camp, 11
Tags: number theory proposed , number theory
Find all integers $ n\ge 2$ with the following property: There exists three distinct primes $p,q,r$ such that whenever $ a_1,a_2,a_3,\cdots,a_n$ are $ n$ distinct positive integers with the property that at least one of $ p,q,r$ divides $ a_j - a_k \ \forall 1\le j\le k\le n$, one of $ p,q,r$ divides all of these differences.

2010 Korea National Olympiad, 1
Tags: modular arithmetic , logarithms , number theory proposed , number theory
Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

1979 IMO Longlists, 5
Tags: floor function , number theory proposed , number theory
Describe which positive integers do not belong to the set \[E = \left\{ \lfloor n+ \sqrt n +\frac 12 \rfloor | n \in \mathbb N\right\}.\]

2008 Indonesia MO, 3
Tags: number theory proposed , number theory
Find all natural number which can be expressed in $ \frac{a\plus{}b}{c}\plus{}\frac{b\plus{}c}{a}\plus{}\frac{c\plus{}a}{b}$ where $ a,b,c\in \mathbb{N}$ satisfy $ \gcd(a,b)\equal{}\gcd(b,c)\equal{}\gcd(c,a)\equal{}1$