Olympiad problems

2016 BAMO, 3

The ${\textit{distinct prime factors}}$ of an integer are its prime factors listed without repetition. For example, the distinct prime factors of $40$ are $2$ and $5$. Let $A=2^k - 2$ and $B= 2^k \cdot A$, where $k$ is an integer ($k \ge 2$). Show that, for every integer $k$ greater than or equal to $2$, [list=i] [*] $A$ and $B$ have the same set of distinct prime factors. [*] $A+1$ and $B+1$ have the same set of distinct prime factors. [/list]

2016 Lusophon Mathematical Olympiad, 1

Tags: number theory , Prime factor , Product , coprime , positive integers

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

2018 German National Olympiad, 5

Tags: number theory , Sequence , Prime factor , primes

We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.

2008 China Northern MO, 3

Tags: number theory , Prime factor , divides

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.