Olympiad problems

1990 Swedish Mathematical Competition, 1

Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.

2014 IFYM, Sozopol, 2

Tags: number theory , divisor

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

1986 Tournament Of Towns, (129) 4

Tags: number theory , divisible , factorial , divisor , divide

We define $N !!$ to be $N(N - 2)(N -4)...5 \cdot 3 \cdot 1$ if $N$ is odd and $N(N -2)(N -4)... 6\cdot 4\cdot 2$ if $N$ is even . For example, $8 !! = 8 \cdot 6\cdot 4\cdot 2$ , and $9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1$ . Prove that $1986 !! + 1985 !!$ i s divisible by $1987$. (V.V . Proizvolov , Moscow)

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]

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: divisor , number of divisors , number theory

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 $$(a_1+a_2+...+a_n)^2={a_1}^3+{a_2}^3+...+{a_n}^3$$

1997 Singapore MO Open, 3

Tags: divisor , number theory , factor , diophantine equation , diophantine

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

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 $3804$.

2005 Germany Team Selection Test, 1

Tags: equation , number theory , 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 Estonia Team Selection Test, 5

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

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 $k$.

2009 Tournament Of Towns, 3

Tags: recurrence relation , divisor , 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 $b$.

2017 International Zhautykov Olympiad, 2

Tags: number theory , prime , divisor

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

1992 Austrian-Polish Competition, 1

Tags: divisor , number theory , sum , positive , product , even

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

2022 All-Russian Olympiad, 1

Tags: divisor , all russia mo , number theory

We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

2013 NZMOC Camp Selection Problems, 12

Tags: number theory , inequalities , prime divisor , divisor , prime

For a positive integer $n$, let $p(n)$ denote the largest prime divisor of $n$. Show that there exist infinitely many positive integers m such that $p(m-1) < p(m) < p(m + 1)$.

2002 IMO, 4

Tags: inequalities , number theory , divisor

Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.

2019 AMC 10, 9

Tags: divisor , divisibility

What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers? $\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

2016 Federal Competition For Advanced Students, P1, 4

Tags: number theory , ratio , divisor

Determine all composite positive integers $n$ with the following property: If $1 = d_1 < d_2 < \cdots < d_k = n$ are all the positive divisors of $n$, then $$(d_2 - d_1) : (d_3 - d_2) : \cdots : (d_k - d_{k-1}) = 1:2: \cdots :(k-1)$$ (Walther Janous)

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divisor

For a positive integer $m$ we denote by $\tau (m)$ the number of its positive divisors, and by $\sigma (m)$ their sum. Determine all positive integers $n$ for which $n \sqrt{ \tau (n) }\le \sigma(n)$

2016 Dutch IMO TST, 2

Tags: number theory , divisible , divisor , exponential

Determine all pairs $(a, b)$ of integers having the following property: there is an integer $d \ge 2$ such that $a^n + b^n + 1$ is divisible by $d$ for all positive integers $n$.

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: number theory , sequence , divisor , number theory with sequences , integer sequence

In the sequence of integers $(a_n)$, the sum $a_m + a_n$ is divided by $m + n$ with any different $m$ and $n$. Prove that $a_n$ is a multiple of $n$ for any $n$.

2012 Danube Mathematical Competition, 4

Tags: sum , subset , set , divisor , combinatorics

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 $\frac{n + 1}{2}$.

1997 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: divisor , number theory

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

2018 Regional Competition For Advanced Students, 4

Tags: number theory , number of divisors , divisor

Let $d(n)$ be the number of all positive divisors of a natural number $n \ge 2$. Determine all natural numbers $n \ge 3$ such that $d(n -1) + d(n) + d(n + 1) \le 8$. [i]Proposed by Richard Henner[/i]

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine , diophantine equation , divisor

Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

2006 Cuba MO, 5

Tags: number theory , prime , divisor

The following sequence of positive integers $a_1, a_2, ..., a_{400}$ satisfies the relationship $a_{n+1} = \tau (a_n) + \tau (n)$ for all $1 \le n \le 399$, where $\tau (k) $ is the number of positive integer divisors that $k$ has. Prove that in the sequence there are no more than $210$ prime numbers.