Olympiad problems

1999 Ukraine Team Selection Test, 10

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 $n \in N$.

2014 Contests, 3

Tags: polynomial , number theory , prime divisor

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 $(n+1)^4 + (n+1)^2 +1$.

1981 Austrian-Polish Competition, 7

Tags: number theory , prime divisor , 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.

2021 Greece JBMO TST, 2

Tags: game , combinatorics , game strategy , 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.

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 $(n+1)^4 + (n+1)^2 +1$.

1986 Austrian-Polish Competition, 7

Tags: factorial , divide , prime divisor

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 $n!$.

2002 Kazakhstan National Olympiad, 7

Tags: prime divisor , divisor , divide , number theory

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

2016 Taiwan TST Round 3, 5

Tags: polynomial , algebra , 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.

2015 Taiwan TST Round 3, 3

Tags: number theory , sequence , prime divisor

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]

2015 Azerbaijan National Olympiad, 4

Tags: number theory , prime divisor , divisor

Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

1970 Czech and Slovak Olympiad III A, 1

Tags: number theory , prime , fraction , harmonic , number , prime divisor

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 $a.$

2024 Bangladesh Mathematical Olympiad, P8

Tags: prime divisor , nt construction , number theory

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]

2013 NZMOC Camp Selection Problems, 12

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

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

2014 IMO Shortlist, N7

Tags: sequence , prime divisor , number theory

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]

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 $n$.

1987 Austrian-Polish Competition, 2

Tags: algebra , polynomial , prime divisor

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 $x = y$.

2024 Regional Olympiad of Mexico Southeast, 1

Tags: number theory , integer pairs , prime divisor , 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.

KoMaL A Problems 2017/2018, A. 720

Tags: set , prime divisor , number theory

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 France Team Selection Test, 3

Tags: polynomial , prime divisor , number theory

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 $(n+1)^4 + (n+1)^2 +1$.

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 $(n+1)^4 + (n+1)^2 +1$.

2000 Estonia National Olympiad, 4

Tags: number theory , sequence , prime divisor

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 $a_{2000} + b_{2000}$.

2005 China Team Selection Test, 1

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

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 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 $$\pi \left(q^{2^k}-1\right)< \pi\left(q^{2^k}\right)<\pi \left( q^{2^k}+1\right).$$