This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 6

2016 Lusophon Mathematical Olympiad, 1
Tags: positive integer , prime factor , product , number theory , coprime
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?

2016 BAMO, 3
Tags: prime factorization , prime factor , factor , number theory
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]

2023 Argentina National Olympiad, 2
Tags: prime factor , number theory
Find all positive integers $n$ such that all prime factors of $2^n-1$ are less than or equal to $7$.

2017 Macedonia JBMO TST, 5
Tags: number theory , digit , exponent , prime factor
Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

2008 China Northern MO, 3
Tags: number theory , prime factor , divide
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 $n!$.

2018 German National Olympiad, 5
Tags: number theory , sequence , prime factor , prime
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.