Olympiad problems

2011 Korea Junior Math Olympiad, 6

For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.

2016 EGMO, 6

Tags: divisor , number theory

Let $S$ be the set of all positive integers $n$ such that $n^4$ has a divisor in the range $n^2 +1, n^2 + 2,...,n^2 + 2n$. Prove that there are infinitely many elements of $S$ of each of the forms $7m, 7m+1, 7m+2, 7m+5, 7m+6$ and no elements of $S$ of the form $7m+3$ and $7m+4$, where $m$ is an integer.

2004 Estonia Team Selection Test, 5

Tags: lcm , power of 2 , divisor , number theory

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2016 Switzerland Team Selection Test, Problem 7

Tags: divisor , number theory

Find all positive integers $n$ such that $$\sum_{d|n, 1\leq d <n}d^2=5(n+1)$$

2021 Ukraine National Mathematical Olympiad, 8

Tags: divisor , number theory

Given a natural number $n$. Prove that you can choose $ \phi (n)+1 $ (not necessarily different) divisors $n$ with the sum $n$. Here $ \phi (n)$ denotes the number of natural numbers less than $n$ that are coprime with $n$. (Fedir Yudin)

2005 India IMO Training Camp, 2

Tags: equation , divisor , number theory

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

1995 All-Russian Olympiad Regional Round, 9.5

Tags: divisor , number theory

Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).

2004 Estonia Team Selection Test, 5

Tags: lcm , power of 2 , divisor , number theory

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2012 Tournament of Towns, 2

Tags: number theory , divisor

The number $4$ has an odd number of odd positive divisors, namely $1$, and an even number of even positive divisors, namely $2$ and $4$. Is there a number with an odd number of even positive divisors and an even number of odd positive divisors?

2020 AIME Problems, 4

Tags: divisor

Let $S$ be the set of positive integers $N$ with the property that the last four digits of $N$ are $2020$, and when the last four digits are removed, the result is a divisor of $N$. For example, $42,020$ is in $S$ because $4$ is a divisor of $42,020$. Find the sum of all the digits of all the numbers in $S$. For example, the number $42,020$ contributes $4+2+0+2+0=8$ to this total.

2005 AIME Problems, 3

Tags: divisor

How many positive integers have exactly three proper divisors, each of which is less than 50?

2014 Federal Competition For Advanced Students, P2, 1

Tags: number of divisors , divisor , number theory

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2018 Singapore Junior Math Olympiad, 4

Tags: factor , sum , divisor , number theory

Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors.

2009 Tournament Of Towns, 3

Tags: divisor , recurrence relation , number theory

For each positive integer $n$, denote by $O(n)$ its greatest odd divisor. Given any positive integers $x_1 = a$ and $x_2 = b$, construct an innite sequence of positive integers as follows: $x_n = O(x_{n-1} + x_{n-2})$, where $n = 3,4,...$ (a) Prove that starting from some place, all terms of the sequence are equal to the same integer. (b) Express this integer in terms of $a$ and $b$.

2025 Kyiv City MO Round 2, Problem 3

Tags: number theory , divisor

A positive integer $ n $, which has at least one proper divisor, is divisible by the arithmetic mean of the smallest and largest of its proper divisors (which may coincide). What can be the number of divisors of $ n $? [i]A proper divisor of a positive integer $ n $ is any of its divisors other than $ 1 $ and $ n $.[/i] [i]Proposed by Mykhailo Shtandenko[/i]

2020 Malaysia IMONST 1, 17

Tags: divisor , number theory

Given a positive integer $n$. The number $2n$ has $28$ positive factors, while the number $3n$ has $30$ positive factors. Find the number of positive divisors of $6n$.

2014 EGMO, 3

Tags: quadratic , combinatorial number theory , divisor , number theory

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

2019 Germany Team Selection Test, 1

Tags: arithmetic function , number of divisors , divisor , number theory

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2005 USAMO, 1

Tags: arrangement , induction , prime number , relatively prime , combinatorics , divisor , number theory

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

2006 Junior Balkan Team Selection Tests - Romania, 3

Tags: sum of digits , divisor , number theory

For any positive integer $n$ let $s(n)$ be the sum of its digits in decimal representation. Find all numbers $n$ for which $s(n)$ is the largest proper divisor of $n$.

2021 Thailand TSTST, 1

Tags: prime factorization , divisor , number theory

For each positive integer $n$, let $\rho(n)$ be the number of positive divisors of $n$ with exactly the same set of prime divisors as $n$. Show that, for any positive integer $m$, there exists a positive integer $n$ such that $\rho(202^n+1)\geq m.$

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: divisor , perfect square , number theory

Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

2024 Romania EGMO TST, P4

Tags: divisor , number theory

Find all composite positive integers $a{}$ for which there exists a positive integer $b\geqslant a$ with the same number of divisors as $a{}$ with the following property: if $a_1<\cdots<a_n$ and $b_1<\cdots<b_n$ are the proper divisors of $a{}$ and $b{}$ respectively, then $a_i+b_i, 1\leqslant i\leqslant n$ are the proper divisors of some positive integer $c.{}$

1995 May Olympiad, 1

Tags: game , game strategy , divisor , number theory

Veronica, Ana and Gabriela are forming a round and have fun with the following game. One of them chooses a number and says out loud, the one to its left divides it by its largest prime divisor and says the result out loud and so on. The one who says the number out loud $1$ wins , at which point the game ends. Ana chose a number greater than $50$ and less than $100$ and won. Veronica chose the number following the one chosen by Ana and also won. Determine all the numbers that could have been chosen by Ana.

BIMO 2021, 1

Tags: perfect number , number theory , divisor

Given a natural number $n$, call a divisor $d$ of $n$ to be $\textit{nontrivial}$ if $d>1$. A natural number $n$ is $\textit{good}$ if one or more distinct nontrivial divisors of $n$ sum up to $n-1$. Prove that every natural number $n$ has a multiple that is good.