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: 48

2024 Bulgarian Autumn Math Competition, 10.3
Tags: number theory , polynomial , prime divisor , floor function , algebra
Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.

2024 Regional Olympiad of Mexico Southeast, 1
Tags: integer pairs , prime divisor , number theory , prime
Find all pairs of positive integers \(a, b\) such that the numbers \(a+1\), \(b+1\), \(2a+1\), \(2b+1\), \(a+3b\), and \(b+3a\) are all prime numbers.

2014 France Team Selection Test, 3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2015 Latvia Baltic Way TST, 15
Tags: prime divisor , number theory , divisor
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 $w(n) < w(n + 1) < w(n + 2)$.

1999 Ukraine Team Selection Test, 10
Tags: inequalities , min , prime divisor , number theory
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 $n \in N$.

2021 Greece JBMO TST, 2
Tags: game , game strategy , combinatorics , winning strategy , prime divisor
Anna and Basilis play a game writing numbers on a board as follows: The two players play in turns and if in the board is written the positive integer $n$, the player whose turn is chooses a prime divisor $p$ of $n$ and writes the numbers $n+p$. In the board, is written at the start number $2$ and Anna plays first. The game is won by whom who shall be first able to write a number bigger or equal to $31$. Find who player has a winning strategy, that is who may writing the appropriate numbers may win the game no matter how the other player plays.

2013 IMO Shortlist, N3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1995 Brazil National Olympiad, 3
Tags: greatest common divisor , number theory , prime divisor , inequalities
For any positive integer $ n>1$, let $ P\left(n\right)$ denote the largest prime divisor of $ n$. Prove that there exist infinitely many positive integers $ n$ for which \[ P\left(n\right)<P\left(n\plus{}1\right)<P\left(n\plus{}2\right).\]

VMEO IV 2015, 12.2
Tags: number theory , sum of divisors , prime divisor
Given a positive integer $k$. Prove that there are infinitely many positive integers $n$ satisfy the following conditions at the same time: a) $n$ has at least $k$ distinct prime divisors b) All prime divisors other than $3$ of $n$ have the form $4t+1$, with $t$ some positive integer. c) $n | 2^{\sigma(n)}-1$ Here $\sigma(n)$ demotes the sum of the positive integer divisors of $n$.

2014 Taiwan TST Round 1, 5
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

1970 Czech and Slovak Olympiad III A, 1
Tags: number theory , prime , fraction , harmonic , prime divisor , number
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 $a.$

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.

2013 NZMOC Camp Selection Problems, 12
Tags: prime divisor , 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 $p(m-1) < p(m) < p(m + 1)$.

2008 IMO Shortlist, 6
Tags: quadratic , number theory , prime divisor
Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$. [i]Author: Kestutis Cesnavicius, Lithuania[/i]

2014 Contests, 3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2024 Bangladesh Mathematical Olympiad, P8
Tags: number theory , prime divisor , nt construction
Let $k$ be a positive integer. Show that there exist infinitely many positive integers $n$ such that $\frac{n^n-1}{n-1}$ has at least $k$ distinct prime divisors. [i]Proposed by Adnan Sadik[/i]

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)

2018 Saudi Arabia IMO TST, 1
Tags: prime divisor , theory , divisor , prime
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).

2016 Postal Coaching, 2
Tags: number theory , prime divisor
Let $\pi (n)$ denote the largest prime divisor of $n$ for any positive integer $n > 1$. Let $q$ be an odd prime. Show that there exists a positive integer $k$ such that $$\pi \left(q^{2^k}-1\right)< \pi\left(q^{2^k}\right)<\pi \left( q^{2^k}+1\right).$$

KoMaL A Problems 2017/2018, A. 720
Tags: number theory , prime divisor , set
We call a positive integer [i]lively[/i] if it has a prime divisor greater than $10^{10^{100}}$. Prove that if $S$ is an infinite set of lively positive integers, then it has an infinite subset $T$ with the property that the sum of the elements in any finite nonempty subset of $T$ is a lively number.

2014 Belarus Team Selection Test, 3
Tags: number theory , prime divisor , polynomial
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

2015 Azerbaijan National Olympiad, 4
Tags: prime divisor , number theory , divisor
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2005 China Team Selection Test, 1
Tags: prime divisor , number theory , prime number , divisibility , power of 2
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.

2016 Taiwan TST Round 3, 5
Tags: algebra , polynomial , number theory , prime divisor
Let $f(x)$ be the polynomial with integer coefficients ($f(x)$ is not constant) such that \[(x^3+4x^2+4x+3)f(x)=(x^3-2x^2+2x-1)f(x+1)\] Prove that for each positive integer $n\geq8$, $f(n)$ has at least five distinct prime divisors.

1981 Austrian-Polish Competition, 7
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.