Olympiad problems

2009 Tournament Of Towns, 3

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 $b$.

1974 Bundeswettbewerb Mathematik, 4

Tags: number theory , divisor , odd , remainder

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

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 $k$.

2017 Germany, Landesrunde - Grade 11/12, 4

Tags: combinatorics , divisor , number theory

Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors.

2019 AMC 10, 19

Tags: divisor

Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$ $\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$

2009 Switzerland - Final Round, 10

Tags: number theory , divisor

Let $n > 3$ be a natural number. Prove that $4^n + 1$ has a prime divisor $> 20$.

1984 IMO Shortlist, 3

Tags: number theory , equation , divisor

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

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 $d_4+d_5=d_6+7$.

2024 AMC 10, 8

Tags: divisor

Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$? $ \textbf{(A) }0 \qquad \textbf{(B) }2 \qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }8 \qquad $

2022 Switzerland - Final Round, 5

Tags: number theory , recurrence relation , divisor

For an integer $a \ge 2$, denote by $\delta_(a) $ the second largest divisor of $a$. Let $(a_n)_{n\ge 1}$ be a sequence of integers such that $a_1 \ge 2$ and $$a_{n+1} = a_n + \delta_(a_n)$$ for all $n \ge 1$. Prove that there exists a positive integer $k$ such that $a_k$ is divisible by $3^{2022}$.

2012 Tournament of Towns, 2

Tags: number theory , prime , divisor

Let $C(n)$ be the number of prime divisors of a positive integer n. (For example, $C(10) = 2,C(11) = 1, C(12) = 2$). Consider set S of all pairs of positive integers $(a, b)$ such that $a\ne b$ and $C(a + b) = C(a) + C(b)$. Is set $S$ finite or infinite?

1980 Dutch Mathematical Olympiad, 2

Tags: number theory , divisor

Find the product of all divisors of $1980^n$, $n \ge 1$.

2021 Romanian Master of Mathematics Shortlist, N1

Tags: number theory , unit fractions , divisor

Given a positive integer $N$, determine all positive integers $n$, satisfying the following condition: for any list $d_1,d_2,\ldots,d_k$ of (not necessarily distinct) divisors of $n$ such that $\frac{1}{d_1} + \frac{1}{d_2} + \ldots + \frac{1}{d_k} > N$, some of the fractions $\frac{1}{d_1}, \frac{1}{d_2}, \ldots, \frac{1}{d_k}$ add up to exactly $N$.

1998 Belarus Team Selection Test, 3

Tags: number theory , divisor

Let $1=d_1<d_2<d_3<...<d_k=n$ be all different divisors of positive integer $n$ written in ascending order. Determine all $n$ such that $$d_7^2+d_{10}^2=(n/d_{22})^2.$$

1999 Korea Junior Math Olympiad, 6

Tags: number theory , divisor

For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4>m$.

2012 India Regional Mathematical Olympiad, 2

Tags: number theory , divisibility , divide , divisor

Prove that for all positive integers $n$, $169$ divides $21n^2 + 89n + 44$ if $13$ divides $n^2 + 3n + 51$.

2016 Dutch BxMO TST, 1

Tags: number theory , divisor , odd , equation

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 $n = 3t(n) + 5r(n)$.

2014 EGMO, 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 $a + b = n$.

2024 Bulgaria MO Regional Round, 11.3

Tags: number theory , divisor , extremal

A positive integer $n$ is called $\textit{good}$ if $2 \mid \tau(n)$ and if its divisors are $$1=d_1<d_2<\ldots<d_{2k-1}<d_{2k}=n, $$ then $d_{k+1}-d_k=2$ and $d_{k+2}-d_{k-1}=65$. Find the smallest $\textit{good}$ number.

2013 Czech-Polish-Slovak Junior Match, 4

Tags: divide , divisor , number theory , digit

Determine the largest two-digit number $d$ with the following property: for any six-digit number $\overline{aabbcc}$ number $d$ is a divisor of the number $\overline{aabbcc}$ if and only if the number $d$ is a divisor of the corresponding three-digit number $\overline{abc}$. Note The numbers $a \ne 0, b$ and $c$ need not be different.

2007 Peru MO (ONEM), 3

Tags: number theory , consecutive , divisor

We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.

2023 ISL, N1

Tags: number theory , divisor

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

2019 Saint Petersburg Mathematical Olympiad, 4

Tags: number theory , reciprocal sum , sum , irreducible , divisor

Olya wrote fractions of the form $1 / n$ on cards, where $n$ is all possible divisors the numbers $6^{100}$ (including the unit and the number itself). These cards she laid out in some order. After that, she wrote down the number on the first card, then the sum of the numbers on the first and second cards, then the sum of the numbers on the first three cards, etc., finally, the sum of the numbers on all the cards. Every amount Olya recorded on the board in the form of irreducible fraction. What is the least different denominators could be on the numbers on the board?

2002 IMO Shortlist, 2

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 $n^2$.

2024 VJIMC, 4

Tags: integer , function , number theory , sequence , divisor

Let $(b_n)_{n \ge 0}$ be a sequence of positive integers satisfying $b_n=d\left(\sum_{i=0}^{n-1} b_k\right)$ for all $n \ge 1$. (By $d(m)$ we denote the number of positive divisors of $m$.) a) Prove that $(b_n)_{n \ge 0}$ is unbounded. b) Prove that there are infinitely many $n$ such that $b_n>b_{n+1}$.