Olympiad problems

1992 ITAMO, 3

Prove that for each $n \ge 3$ there exist $n$ distinct positive divisors $d_1,d_2, ...,d_n$ of $n!$ such that $n! = d_1 +d_2 +...+d_n$.

2021 Czech and Slovak Olympiad III A, 4

Tags: number theory , divisor , divide

Find all natural numbers $n$ for which equality holds $n + d (n) + d (d (n)) +... = 2021$, where $d (0) = d (1) = 0$ and for $k> 1$, $ d (k)$ is the [i]superdivisor [/i] of the number $k$ (i.e. its largest divisor of $d$ with property $d <k$). (Tomáš Bárta)

2006 Portugal MO, 5

Tags: number theory , divisor

Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.

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 $n$.

The Golden Digits 2024, P1

Tags: number theory , divisor

Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$? [i]Proposed by Pavel Ciurea[/i]

2016 India Regional Mathematical Olympiad, 3

Tags: number theory , divide , divisor

$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , number theory , difference , divisor

Let $n$ be a positive integer. For each of the numbers $1, 2,.., n$ we compute the difference between the number of its odd positive divisors and its even positive divisors. Prove that the sum of these differences is at least $0$ and at most $n$.

2011 Korea Junior Math Olympiad, 6

Tags: number theory , euler s totient function , relatively prime , coprime , divisor

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 $p|q^2 - 1$.

2019 Regional Competition For Advanced Students, 4

Tags: number theory , divisor

Find all natural numbers $n$ that are smaller than $128^{97}$ and have exactly $2019$ divisors.

2020 Dürer Math Competition (First Round), P1

Tags: number theory , divisor

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?

1986 IMO Longlists, 58

Tags: number theory , prime factorization , divisor

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors.

2016 Saudi Arabia GMO TST, 3

Tags: divide , divisor , polynomial , algebra

Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

2024 Dutch IMO TST, 1

Tags: coprime , number theory , divisor

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 $\alpha(n)=\beta(n)$.

1999 All-Russian Olympiad Regional Round, 9.7

Tags: number theory , divisor

Prove that every natural number is the difference of two natural numbers that have the same number of prime factors. (Each prime divisor is counted once, for example, the number $12$ has two prime factors: $2$ and $3$.)

2001 Estonia National Olympiad, 4

Tags: number theory , harmonic , divisor

We call a triple of positive integers $(a, b, c)$ [i]harmonic [/i] if $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$. Prove that, for any given positive integer $c$, the number of harmonic triples $(a, b, c)$ is equal to the number of positive divisors of $c^2$.

2015 Postal Coaching, Problem 4

Tags: number theory , divisor

For $ n \in \mathbb{N}$, let $s(n)$ denote the sum of all positive divisors of $n$. Show that for any $n > 1$, the product $s(n - 1)s(n)s(n + 1)$ is an even number.

2018 Saudi Arabia BMO TST, 3

Tags: divide , divisor , number theory

Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

2022 Saudi Arabia BMO + EGMO TST, 2.2

Tags: number theory , divisor

Find all positive integers $n$ that have precisely $\sqrt{n + 1}$ natural divisors.

1983 IMO Shortlist, 2

Tags: number theory , sum of divisors , inequality , divisor

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2003 France Team Selection Test, 3

Tags: algebra , number theory , divisor , prime

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2015 Dutch BxMO/EGMO TST, 1

Tags: number theory , divisor , divide

Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

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.

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: number theory , number of divisors , divisor

It is given positive integer $N$. Let $d_1$, $d_2$,...,$d_n$ be its divisors and let $a_i$ be number of divisors of $d_i$, $i=1,2,...n$. Prove that $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

2018 Poland - Second Round, 2

Tags: number theory , combinatorics , inequalities , divisor

Let $n$ be a positive integer, which gives remainder $4$ of dividing by $8$. Numbers $1 = k_1 < k_2 < ... < k_m = n$ are all positive diivisors of $n$. Show that if $i \in \{ 1, 2, ..., m - 1 \}$ isn't divisible by $3$, then $k_{i + 1} \le 2k_{i}$.

2019 China Team Selection Test, 2

Tags: divisor , number theory

Let $S$ be a set of positive integers, such that $n \in S$ if and only if $$\sum_{d|n,d<n,d \in S} d \le n$$ Find all positive integers $n=2^k \cdot p$ where $k$ is a non-negative integer and $p$ is an odd prime, such that $$\sum_{d|n,d<n,d \in S} d = n$$