Olympiad problems

1991 Bundeswettbewerb Mathematik, 2

Let $g$ be an even positive integer and $f(n) = g^n + 1$ , $(n \in N^* )$. Prove that for every positive integer $n$ we have: a) $f(n)$ divides each of the numbers $f(3n), f(5n), f(7n)$ b) $f(n)$ is relative prime to each of the numbers $f(2n), f(4n),f(6n),...$

2022 New Zealand MO, 5

Tags: number theory , multiple , divide , divisible , recurrence relation

The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.

1993 All-Russian Olympiad Regional Round, 11.2

Tags: number theory , divide , divisible

Prove that, for every integer $n > 2$, the number $$\left[\left( \sqrt[3]{n}+\sqrt[3]{n+2}\right)^3\right]+1$$ is divisible by $8$.

2005 Thailand Mathematical Olympiad, 11

Tags: number theory , divide

Find the smallest positive integer $x$ such that $2^{254}$ divides $x^{2005} + 1$.

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)

2010 Grand Duchy of Lithuania, 5

Tags: divide , number theory

Find positive integers n that satisfy the following two conditions: (a) the quotient obtained when $n$ is divided by $9$ is a positive three digit number, that has equal digits. (b) the quotient obtained when $n + 36$ is divided by $4$ is a four digit number, the digits beeing $2, 0, 0, 9$ in some order.

2020 Argentina National Olympiad, 1

Tags: number theory , divide , sum of digits

For every positive integer $n$, let $S (n)$ be the sum of the digits of $n$. Find, if any, a $171$-digit positive integer $n$ such that $7$ divides $S (n)$ and $7$ divides $S (n + 1)$.

1991 Chile National Olympiad, 4

Tags: number theory , divide , divisible

Show that the expressions $2x + 3y$, $9x + 5y$ are both divisible by $17$, for the same values of $x$ and $y$.

2012 Brazil Team Selection Test, 2

Tags: number theory , divide , divisible

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

2016 Dutch Mathematical Olympiad, 3

Tags: greatest common divisor , gcd , number theory , divisor , divide

Find all possible triples $(a, b, c)$ of positive integers with the following properties: • $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$, • $a$ is a divisor of $a + b + c$, • $b$ is a divisor of $a + b + c$, • $c$ is a divisor of $a + b + c$. (Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)

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.

2017 May Olympiad, 5

Tags: number theory , divide , divisible

We will say that two positive integers $a$ and $b$ form a [i]suitable pair[/i] if $a+b$ divides $ab$ (its sum divides its multiplication). Find $24$ positive integers that can be distribute into $12$ suitable pairs, and so that each integer number appears in only one pair and the largest of the $24$ numbers is as small as possible.

2020 Swedish Mathematical Competition, 1

Tags: number theory , divide , divisible

How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?

2018 Saudi Arabia GMO TST, 2

Tags: number theory , divide , divisible , prime

Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.

2015 Balkan MO Shortlist, N3

Tags: prime divisor , integer , number theory , divide , divisor

Let $a$ be a positive integer. For all positive integer n, we define $ a_n=1+a+a^2+\ldots+a^{n-1}. $ Let $s,t$ be two different positive integers with the following property: If $p$ is prime divisor of $s-t$, then $p$ divides $a-1$. Prove that number $\frac{a_{s}-a_{t}}{s-t}$ is an integer. (FYROM)

2007 Estonia National Olympiad, 1

Tags: divide , number theory , digit

The seven-digit integer numbers are different in pairs and this number is divided by each of its own numbers. a) Find all possibilities for the three numbers that are not included in this number. b) Give an example of such a number.

2016 Saudi Arabia Pre-TST, 1.4

Tags: number theory , coprime , divide , divisible

Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

2006 May Olympiad, 1

Tags: number theory , divide

Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

2000 Tournament Of Towns, 2

Tags: number theory , divide , divisible

Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$. (A Spivak)

2022 Federal Competition For Advanced Students, P2, 1

Tags: function , number theory , divide

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

2011 QEDMO 9th, 4

Tags: factorial , number theory , multiple , divide

Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

2021 Regional Competition For Advanced Students, 4

Tags: number theory , divide , divisible

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2006 Thailand Mathematical Olympiad, 5

Tags: divide , divisible , number theory

Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$

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 $D$.

1983 Tournament Of Towns, (038) A5

Tags: number theory , divisible , divide , combinatorics

Prove that in any set of $17$ distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other. (An established theorem)