Olympiad problems

1981 Austrian-Polish Competition, 7

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.

2016 Argentina National Olympiad, 5

Tags: number theory , prime divisors , divisor , algebra

Let $a$ and $b$ be rational numbers such that $a+b=a^2+b^2$ . Suppose the common value $s=a+b=a^2+b^2$ is not an integer, and let's write it as an irreducible fraction: $s=\frac{m}{n}$. Let $p$ be the smallest prime divisor of $n$. Find the minimum value of $p$.

2015 Latvia Baltic Way TST, 15

Tags: number theory , Divisors , prime divisor , prime divisors

Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.

2009 Bundeswettbewerb Mathematik, 2

Tags: number theory , quadratic trinomial , trinomial , prime divisors , prime , Divisors

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.

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , prime divisors , Fractions , harmonic , numbers

Let $p>2$ be a prime and $a,b$ positive integers such that \[\frac ab=1+\frac12+\frac13+\cdots+\frac{1}{p-1}.\] Show that $p$ is a divisor of $a.$

2018 Saudi Arabia IMO TST, 1

Tags: prime divisors , Divisors , primes , theory

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

2023 Romania EGMO TST, P2

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

1986 Austrian-Polish Competition, 7

Tags: factorial , divides , prime divisors

Let $k$ and $n$ be integers with $0 < k < n^2/4$ such that k has no prime divisor greater than $n$. Prove that $k$ divides $n!$.

2015 Peru IMO TST, 16

Tags: IMO Shortlist , number theory , Sequence , prime divisors

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

2014 IMO Shortlist, N7

Tags: IMO Shortlist , number theory , Sequence , prime divisors

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

1997 IMO Shortlist, 14

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2015 IFYM, Sozopol, 5

Tags: number theory , prime divisors , prime numbers , Divisibility

Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

2013 NZMOC Camp Selection Problems, 12

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

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 $p(m-1) < p(m) < p(m + 1)$.

2007 Indonesia TST, 2

Tags: prime divisors , Divisors , number theory

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 Estonia National Olympiad, 4

Tags: Sequence , prime divisors , number theory

Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions $a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$. Find all different prime factors οf the number $a_{2000} + b_{2000}$.

2022 Macedonian Mathematical Olympiad, Problem 3

Tags: Sequence , prime divisors , number theory

The sequence $(a_n)_{n \ge 1}^\infty$ is given by: $a_1=2$ and $a_{n+1}=a_n^2+a_n$ for all $n \ge 1$. For an integer $m \ge 2$, $L(m)$ denotes the greatest prime divisor of $m$. Prove that there exists some $k$, for which $L(a_k) > 1000^{1000}$. [i]Proposed by Nikola Velov[/i]

1987 Austrian-Polish Competition, 2

Tags: polynomial , algebra , prime divisors

Let $n$ be the square of an integer whose each prime divisor has an even number of decimal digits. Consider $P(x) = x^n - 1987x$. Show that if $x,y$ are rational numbers with $P(x) = P(y)$, then $x = y$.

1999 Ukraine Team Selection Test, 10

Tags: number theory , prime divisors , inequalities , min

For a natural number $n$, let $w(n)$ denote the number of (positive) prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{w(n)} \le k \sqrt[4]{ n}$ for each $n \in N$.

2015 Taiwan TST Round 3, 3

Tags: IMO Shortlist , number theory , Sequence , prime divisors

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

2005 China Team Selection Test, 1

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2007 Indonesia TST, 2

Tags: prime divisors , Divisors , number theory

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.