Olympiad problems

2024 Francophone Mathematical Olympiad, 4

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2001 Estonia National Olympiad, 2

Tags: least common multiple , divide , divisor , number theory

A student wrote a correct addition operation $A/B+C/D = E/F$ on the blackboard, where both summands are irreducible and $F$ is the least common multiple of $B$ and $D$. After that, the student reduced the sum $E/F$ correctly by an integer $d$. Prove that $d$ is a common divisor of $B$ and $D$.

1956 Moscow Mathematical Olympiad, 323

Tags: number theory , divisor , divide

a) Find all integers that can divide both the numerator and denominator of the ratio $\frac{5m + 6}{8m + 7}$ for an integer $m$. b) Let $a, b, c, d, m$ be integers. Prove that if the numerator and denominator of the ratio $\frac{am + b}{cm+ d}$ are both divisible by $k$, then so is $ad - bc$.

2016 Federal Competition For Advanced Students, P1, 4

Tags: ratio , number theory , 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)

1982 Bundeswettbewerb Mathematik, 1

Tags: divisor , decimal representation , algebra

Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.

1988 Mexico National Olympiad, 2

Tags: number theory , divisor

If $a$ and $b$ are positive integers, prove that $11a+2b$ is a multiple of $19$ if and only if so is $18a+5b$ .

2024 Belarusian National Olympiad, 10.1

Tags: number theory , divisor

Let $1=d_1<d_2<\ldots<d_k=n$ be all divisors of $n$. It turned out that numbers $d_2-d_1,\ldots,d_k-d_{k-1}$ are $1,3,\ldots,2k-3$ in some order. Find all possible values of $n$ [i]M. Zorka[/i]

2018 Turkey Junior National Olympiad, 1

Tags: number theory , divisor

Let $s(n)$ be the number of positive integer divisors of $n$. Find the all positive values of $k$ that is providing $k=s(a)=s(b)=s(2a+3b)$.

2019 Brazil 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.

2013 Austria Beginners' Competition, 1

Tags: number theory , divisor

Find all natural numbers $n> 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$. (R. Henner, Vienna)

2014 Federal Competition For Advanced Students, P2, 4

Tags: number theory , sum of squares , product , divisor , consecutive

For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?

2024 Baltic Way, 18

Tags: sequence , prime number , bound , number theory , divisor

An infinite sequence $a_1, a_2,\ldots$ of positive integers is such that $a_n \geq 2$ and $a_{n+2}$ divides $a_{n+1} + a_n$ for all $n \geq 1$. Prove that there exists a prime which divides infinitely many terms of the sequence.

Russian TST 2019, P1

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.

2013 IMO Shortlist, N7

Tags: algebra , divisor , number theory

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if \[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good. Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.

2014 Federal Competition For Advanced Students, P2, 1

Tags: number of divisors , divisor , number theory

For each positive natural number $n$ let $d (n)$ be the number of its divisors including $1$ and $n$. For which positive natural numbers $n$, for every divisor $t$ of $n$, that $d (t)$ is a divisor of $d (n)$?

2021 Dutch Mathematical Olympiad, 5

Tags: number theory , prime , divisor

We consider an integer $n > 1$ with the following property: for every positive divisor $d$ of $n$ we have that $d + 1$ is a divisor of$ n + 1$. Prove that $n$ is a prime number.

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 $n$, $s(n) = (c(n) + 1) \cdot \left( \frac{n}{c(n)}+1\right)$

2022 Dutch Mathematical Olympiad, 1

Tags: number theory , divisor

A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

2005 Germany Team Selection Test, 1

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 $n$.

2007 Indonesia TST, 2

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.

2000 Portugal MO, 3

Tags: number theory , divide , divisor

Determine, for each positive integer $n$, the largest positive integer $k$ such that $2^k$ is a divisor of $3^n+1$.

2019 Philippine TST, 3

Tags: number theory , divisor , bounding , infinite sequence

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

2009 Bundeswettbewerb Mathematik, 2

Tags: prime divisor , number theory , quadratic trinomial , trinomial , prime , divisor

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

2005 USAMO, 1

Tags: induction , number theory , prime number , relatively prime , combinatorics , arrangement , divisor

Determine all composite positive integers $n$ for which it is possible to arrange all divisors of $n$ that are greater than 1 in a circle so that no two adjacent divisors are relatively prime.

1998 Switzerland Team Selection Test, 6

Tags: number theory , divisor , prime

Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.