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

2016 Mathematical Talent Reward Programme, SAQ: P 3
Tags: number theory
Prove that for any positive integer $n$ there are $n$ consecutive composite numbers all less than $4^{n+2}$.

1985 Kurschak Competition, 2
Tags: number theory
For every $n\in\mathbb{N}$, define the [i]power sum[/i] of $n$ as follows. For every prime divisor $p$ of $n$, consider the largest positive integer $k$ for which $p^k\le n$, and sum up all the $p^k$'s. (For instance, the power sum of $100$ is $2^6+5^2=89$.) Prove that the [i]power sum[/i] of $n$ is larger than $n$ for infinitely many positive integers $n$.

1998 Junior Balkan MO, 4
Tags: residue , number theory
Do there exist 16 three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by 16? [i]Bulgaria[/i]

2004 Baltic Way, 7
Tags: ratio , number theory
Find all sets $X$ consisting of at least two positive integers such that for every two elements $m,n\in X$, where $n>m$, there exists an element $k\in X$ such that $n=mk^2$.

2006 Tournament of Towns, 6
Tags: decreasing , sequence , number theory
Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)

2017 Dutch IMO TST, 3
Tags: number theory
Let $k > 2$ be an integer. A positive integer $l$ is said to be $k-pable$ if the numbers $1, 3, 5, . . . , 2k - 1$ can be partitioned into two subsets $A$ and $B$ in such a way that the sum of the elements of $A$ is exactly $l$ times as large as the sum of the elements of $B$. Show that the smallest $k-pable$ integer is coprime to $k$.

2022 Federal Competition For Advanced Students, P1, 4
Tags: diophantine , number theory , prime number , diophantine equation
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2021 Indonesia TST, N
Tags: number theory
A positive integer $n$ is said to be $interesting$ if there exist some coprime positive integers $a$ and $b$ such that $n = a^2 - ab + b^2$. Show that if $n^2$ is $interesting$, then $n$ or $3n$ is $interesting$.

2019 IFYM, Sozopol, 2
Tags: functional equation , number theory
Does there exist a strictly increasing function $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for $\forall$ $n\in \mathbb{N}$: $f(f(f(n)))=n+2f(n)$?

2020 Dürer Math Competition (First Round), P1
Tags: divisor , number theory
a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?

2022 Argentina National Olympiad, 2
Tags: number theory
Determine all positive integers $n$ such that numbers from $1$ to $n$ can be sorted in some order $x_1,x_2,...,x_n$ with the property that the number $x_1+x_2+...+x_k$ is divisible by $k$, for all $1\le k\le n$., that is $1$ is divides $x_1$, $2$ divides $x_1+x_2$, $3$ divides $x_1+x_2+x_3$, and so on until $n$ divides $x_1+x_2+...+x_n$.

2012 Kyrgyzstan National Olympiad, 1
Tags: number theory
Prove that $ n $ must be prime in order to have only one solution to the equation $\frac{1}{x}-\frac{1}{y}=\frac{1}{n}$, $x,y\in\mathbb{N}$.

2013 Puerto Rico Team Selection Test, 3
Tags: number theory , prime number
Find all pairs of natural numbers n and prime numbers p such that $\sqrt{n+\frac{p}{n}}$ is a natural number.

2007 District Olympiad, 3
Tags: algebra , functional equation , function , number theory
Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation: $$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$

2014 Balkan MO, 2
Tags: number theory
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that i) there are infinitely many special numbers; ii) $2014$ is not a special number. [i]Romania[/i]

1992 Cono Sur Olympiad, 1
Tags: number theory
Find a positive integrer number $n$ such that, if yor put a number $2$ on the left and a number $1$ on the right, the new number is equal to $33n$.

2019 Saudi Arabia JBMO TST, 1
Tags: number theory
All integer numbers are colored in 3 colors in arbitrary way. Prove that there are two distinct numbers whose difference is a perfect square and the numbers are colored in the same color.

2018 Canada National Olympiad, 5
Tags: number theory
Let $k$ be a given even positive integer. Sarah first picks a positive integer $N$ greater than $1$ and proceeds to alter it as follows: every minute, she chooses a prime divisor $p$ of the current value of $N$, and multiplies the current $N$ by $p^k -p^{-1}$ to produce the next value of $N$. Prove that there are infinitely many even positive integers $k$ such that, no matter what choices Sarah makes, her number $N$ will at some point be divisible by $2018$.

2014 Romania Team Selection Test, 3
Tags: number theory
Determine all positive integers $n$ such that all positive integers less than $n$ and coprime to $n$ are powers of primes.

2009 China Team Selection Test, 1
Tags: number theory
Let $ n$ be a composite. Prove that there exists positive integer $ m$ satisfying $ m|n, m\le\sqrt {n},$ and $ d(n)\le d^3(m).$ Where $ d(k)$ denotes the number of positive divisors of positive integer $ k.$

2010 Contests, 1
Tags: inequalities , number theory
Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

1997 All-Russian Olympiad Regional Round, 9.2
Tags: number theory , combinatorics
The numbers $1, 2, 3, ..., 1000$ are written on the board. Two people take turns erasing one number at a time. The game ends when two numbers remain on the board. If their sum is divisible by three, then the one who made the first move wins. if not, then his partner. Which one will win if played correctly?

2023 Belarusian National Olympiad, 10.2
Tags: number theory , combinatorics
A positive integers has exactly $81$ divisors, which are located in a $9 \times 9$ table such that for any two numbers in the same row or column one of them is divisible by the other one. Find the maximum possible number of distinct prime divisors of $n$

Russian TST 2021, P1
Tags: number theory , representation
Do there exist infinitely many positive integers not expressible in the form \[(a+b)+\log_2(b+c)-2^{c+a},\]where $a,b,c$ are positive integers?

2014 ELMO Shortlist, 7
Tags: number theory
Find all triples $(a,b,c)$ of positive integers such that if $n$ is not divisible by any prime less than $2014$, then $n+c$ divides $a^n+b^n+n$. [i]Proposed by Evan Chen[/i]