Olympiad problems

2015 Azerbaijan National Olympiad, 4

Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2013 Regional Competition For Advanced Students, 1

Tags: probability , number theory , divisor

For which integers between $2000$ and $2010$ (including) is the probability that a random divisor is smaller or equal $45$ the largest?

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 $n$.

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: product , number theory , divisor

Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?

1985 Poland - Second Round, 2

Tags: number theory , sum , divisor

Prove that for a natural number $ n > 2 $ the number $ n! $ is the sum of its $ n $ various divisors.

2022 Durer Math Competition Finals, 16

Tags: number theory , divisor

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

2018 Danube Mathematical Competition, 1

Tags: number theory , composite , divisor

Find all the pairs $(n, m)$ of positive integers which fulfil simultaneously the conditions: i) the number $n$ is composite; ii) if the numbers $d_1, d_2, ..., d_k, k \in N^*$ are all the proper divisors of $n$, then the numbers $d_1 + 1, d_2 + 1, . . . , d_k + 1$ are all the proper divisors of $m$.

2012 QEDMO 11th, 12

Tags: number theory , divide , divisible , divisor

Prove that there are infinitely many different natural numbers of the form $k^2 + 1$, $k \in N$ that have no real divisor of this form.

1981 Austrian-Polish Competition, 7

Tags: prime divisor , number theory , divisor

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

2022 China Team Selection Test, 5

Tags: binomial coefficient , number theory , divisor

Given a positive integer $n$, let $D$ is the set of positive divisors of $n$, and let $f: D \to \mathbb{Z}$ be a function. Prove that the following are equivalent: (a) For any positive divisor $m$ of $n$, \[ n ~\Big|~ \sum_{d|m} f(d) \binom{n/d}{m/d}. \] (b) For any positive divisor $k$ of $n$, \[ k ~\Big|~ \sum_{d|k} f(d). \]

2019 China Team Selection Test, 2

Tags: number theory , divisor

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 $$\sum_{d|n,d<n,d \in S} d = n$$

1989 All Soviet Union Mathematical Olympiad, 490

Tags: number theory , divisor

A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.

2016 EGMO, 6

Tags: number theory , divisor

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.

2024 Alborz Mathematical Olympiad, P1

Tags: number theory , complete residue system , divisor

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari

Mathematical Minds 2024, P1

Tags: number theory , divisor

Find all positive integers $n\geqslant 2$ such that $d_{i+1}/d_i$ is an integer for all $1\leqslant i < k$, where $1=d_1<d_2<\dots <d_k=n$ are all the positive divisors of $n$. [i]Proposed by Pavel Ciurea[/i]

2021 Dutch IMO TST, 4

Tags: number theory , divisor

Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.

2021 Bangladeshi National Mathematical Olympiad, 10

Tags: combinatorics , counting , number theory , divisor

A positive integer $n$ is called [i]nice[/i] if it has at least $3$ proper divisors and it is equal to the sum of its three largest proper divisors. For example, $6$ is [i]nice[/i] because its largest three proper divisors are $3,2,1$ and $6=3+2+1$. Find the number of [i]nice[/i] integers not greater than $3000$.

2005 AIME Problems, 3

Tags: divisor

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

1955 Moscow Mathematical Olympiad, 316

Tags: algebra , polynomial , divisor , integer polynomial , rational

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

2011 Cuba MO, 3

Tags: number theory , divisor

Let $n$ be a positive integer and let $$1 = d_1 < d_2 < d_3 < d_4$$ the four smallest divisors of $n$. Find all$ n$ such that $$n^2 = d_1 + d_2^2+d_3^3 +d_4^4.$$

2009 Estonia Team Selection Test, 6

Tags: number theory , divisor , product

For any positive integer $n$, let $c(n)$ be the largest divisor of $n$ not greater than $\sqrt{n}$ and let $s(n)$ be the least integer $x$ such that $n < x$ and the product $nx$ is divisible by an integer $y$ where $n < y < x$. Prove that, for every $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

2001 All-Russian Olympiad, 4

Tags: number theory , relatively prime , divisor

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

1995 May Olympiad, 1

Tags: number theory , game , game strategy , divisor

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.

1997 German National Olympiad, 2

Tags: inequalities , greatest odd divisor , divisor , number theory

For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$. Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.

1993 Romania Team Selection Test, 2

Tags: algebra , polynomial , divisor

For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.