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: 310

2015 European Mathematical Cup, 3
Tags: number theory , divisor
Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

2003 Estonia National Olympiad, 4
Tags: number theory , sum , reciprocal , divisor
Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

2013 IFYM, Sozopol, 3
Tags: number theory , divisor
The number $A$ is a product of $n$ distinct natural numbers. Prove that $A$ has at least $\frac{n(n-1)}{2}+1$ distinct divisors (including 1 and $A$).

1992 Austrian-Polish Competition, 6
Tags: functional , functional equation , algebra , number theory , divisor
A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2008 IMO Shortlist, 5
Tags: function , number theory , modular arithmetic , divisor , functional equation
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties: [list][*] $ d\left(f(x)\right) \equal{} x$ for all $ x\in\mathbb{N}$. [*] $ f(xy)$ divides $ (x \minus{} 1)y^{xy \minus{} 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.[/list] [i]Proposed by Bruno Le Floch, France[/i]

2012 Tournament of Towns, 4
Tags: prime , divisor , number theory
Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

2011 Saudi Arabia Pre-TST, 4.4
Tags: number theory , divisor , divide
Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

2018 Dutch IMO TST, 2
Tags: number theory , divisor
Find all positive integers $n$, for which there exists a positive integer $k$ such that for every positive divisor $d$ of $n$, the number $d - k$ is also a (not necessarily positive) divisor of $n$.

2012 Danube Mathematical Competition, 4
Tags: combinatorics , sum , subset , set , divisor
Given a positive integer $n$, show that the set $\{1,2,...,n\}$ can be partitioned into $m$ sets, each with the same sum, if and only if m is a divisor of $\frac{n(n + 1)}{2}$ which does not exceed $\frac{n + 1}{2}$.

1987 Mexico National Olympiad, 6
Tags: number theory , divisor
Prove that for every positive integer n the number $(n^3 -n)(5^{8n+4} +3^{4n+2})$ is a multiple of $3804$.