Olympiad problems

2019 Estonia Team Selection Test, 9

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.

2021 Durer Math Competition Finals, 5

Tags: number theory , remainder , divisor

Let $n$ be a positive integer. Show that every divisors of $2n^2 - 1$ gives a different remainder after division by $2n$.

2022 Mexico National Olympiad, 3

Tags: music , divisor

Let $n>1$ be an integer and $d_1<d_2<\dots<d_m$ the list of its positive divisors, including $1$ and $n$. The $m$ instruments of a mathematical orchestra will play a musical piece for $m$ seconds, where the instrument $i$ will play a note of tone $d_i$ during $s_i$ seconds (not necessarily consecutive), where $d_i$ and $s_i$ are positive integers. This piece has "sonority" $S=s_1+s_2+\dots s_n$. A pair of tones $a$ and $b$ are harmonic if $\frac ab$ or $\frac ba$ is an integer. If every instrument plays for at least one second and every pair of notes that sound at the same time are harmonic, show that the maximum sonority achievable is a composite number.

2014 Contests, 3

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

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 Hong Kong TST, 1

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

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.

2016 Greece JBMO TST, 3

Tags: number theory , greatest common divisor , divisor

Positive integer $n$ is such that number $n^2-9$ has exactly $6$ positive divisors. Prove that GCD $(n-3, n+3)=1$

2018 Estonia Team Selection Test, 8

Tags: number theory , divisor

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

2015 Middle European Mathematical Olympiad, 8

Tags: divisor , c.r.t , number theory

Let $n\ge 2$ be an integer. Determine the number of positive integers $m$ such that $m\le n$ and $m^2+1$ is divisible by $n$.

1996 All-Russian Olympiad Regional Round, 9.5

Tags: number theory , sum , divisor

Find all natural numbers that have exactly six divisors whose sum is $3500$.

2015 NZMOC Camp Selection Problems, 8

Tags: divide , divisible , divisor , number theory

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

2019 Germany Team Selection Test, 1

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

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.

1983 Poland - Second Round, 4

Tags: number theory , divisor , sum

Let $ a(k) $ be the largest odd number by which $ k $ is divisible. Prove that $$ \sum_{k=1}^{2^n} a(k) = \frac{1}{3}(4^n+2).$$

2024 Iberoamerican, 1

Tags: number theory , gcd and lcm , divisor

For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \] and determine all pairs of positive integers $(a,b)$ where we have equality case.

2025 Kyiv City MO Round 2, Problem 2

Tags: number theory , divisor

A positive integer $ n $ satisfies the following conditions: [list] [*] The number $ n $ has exactly $ 60 $ divisors: $ 1 = a_1 < a_2 < \cdots < a_{60} = n $; [*] The number $ n+1 $ also has exactly $ 60 $ divisors: $ 1 = b_1 < b_2 < \cdots < b_{60} = n+1 $. [/list] Let $ k $ be the number of indices $ i $ such that $ a_i < b_i $. Find all possible values of $ k $. [i]Note: Such numbers exist, for example, the numbers $ 4388175 $ and $ 4388176 $ both have $ 60 $ divisors.[/i] [i]Proposed by Anton Trygub[/i]

2005 India IMO Training Camp, 2

Tags: number theory , equation , divisor

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 $n$.

2004 Thailand Mathematical Olympiad, 5

Tags: number theory , divisor

Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.

2017 China Team Selection Test, 1

Tags: number theory , modular arithmetic , divisor

Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.

1997 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: number theory , divisor

Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.

1989 Mexico National Olympiad, 2

Tags: number theory , divisor

Find two positive integers $a,b$ such that $a | b^2, b^2 | a^3, a^3 | b^4, b^4 | a^5$, but $a^5$ does not divide $b^6$

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.

2016 NIMO Problems, 1

Tags: remainder , number theory , divisor

Let $m$ be a positive integer less than $2015$. Suppose that the remainder when $2015$ is divided by $m$ is $n$. Compute the largest possible value of $n$. [i] Proposed by Michael Ren [/i]

2008 Korea Junior Math Olympiad, 6

Tags: number theory , sum of powers , sum , divisible , divisor

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

2016 All-Russian Olympiad, 3

Tags: number theory , greatest common divisor , divisor

Alexander has chosen a natural number $N>1$ and has written down in a line,and in increasing order,all his positive divisors $d_1<d_2<\ldots <d_s$ (where $d_1=1$ and $d_s=N$).For each pair of neighbouring numbers,he has found their greater common divisor.The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N-2$.Find all possible values of $N$.

2015 Czech-Polish-Slovak Junior Match, 5

Tags: divisor , number theory

Determine all natural numbers$ n> 1$ with the property: For each divisor $d> 1$ of number $n$, then $d - 1$ is a divisor of $n - 1$.

2016 Dutch BxMO TST, 1

Tags: divisor , odd , equation , number theory

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.