Olympiad problems

2013 Taiwan TST Round 1, 1

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

2025 Euler Olympiad, Round 2, 1

Tags: Euler Olympiad , number theory , Divisors , relatively prime , prime numbers

Let a pair of positive integers $(n, m)$ that are relatively prime be called [i]intertwined[/i] if among any two divisors of $n$ greater than $1$, there exists a divisor of $m$ and among any two divisors of $m$ greater than $1$, there exists a divisor of $n$. For example, pair $(63, 64)$ is intertwined. [b]a)[/b] Find the largest integer $k$ for which there exists an intertwined pair $(n, m)$ such that the product $nm$ is equal to the product of the first $k$ prime numbers. [b]b)[/b] Prove that there does [b]not[/b] exist an intertwined pair $(n, m)$ such that the product $nm$ is the product of $2025$ distinct prime numbers. [b]c)[/b] Prove that there exists an intertwined pair $(n, m)$ such that the number of divisors of $n$ is greater than $2025$. [i]Proposed by Stijn Cambie, Belgium[/i]

2007 Pre-Preparation Course Examination, 19

Tags: function , induction , number theory , prime numbers , number theory unsolved

Find all functions $f : \mathbb N \to \mathbb N$ such that: i) $f^{2000}(m)=f(m)$ for all $m \in \mathbb N$, ii) $f(mn)=\dfrac{f(m)f(n)}{f(\gcd(m,n))}$, for all $m,n\in \mathbb N$, and iii) $f(m)=1$ if and only if $m=1$.

2012 IMO Shortlist, N5

Tags: algebra , number theory , IMO Shortlist , prime numbers , polynomial

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

2024 Chile Classification NMO Seniors, 3

Tags: combinatorics , Chile , prime numbers

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

2020 Macedonian Nationаl Olympiad, 1

Tags: number theory , fermat's little theorem , GCD , Divisibility , prime numbers

Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

2003 Tournament Of Towns, 1

Tags: arithmetic sequence , number theory , relatively prime , prime numbers , number theory proposed

An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?

2007 Bulgarian Autumn Math Competition, Problem 8.3

Tags: primes , square , number theory , prime numbers

Determine all triplets of prime numbers $p<q<r$, such that $p+q=r$ and $(r-p)(q-p)-27p$ is a square.

2016 Polish MO Finals, 1

Tags: algebra , polynomial , number theory , prime numbers , quadratics

Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.

2016 Junior Regional Olympiad - FBH, 3

Tags: number theory , prime numbers

Prove that when dividing a prime number with $30$, remainder is always not a composite number

2013 Korea Junior Math Olympiad, 4

Tags: number theory , prime numbers , minimum , divides

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

2022 Bulgarian Spring Math Competition, Problem 12.4

Tags: number theory , prime numbers , abstract algebra

Let $m$ and $n$ be positive integers and $p$ be a prime number. Find the greatest positive integer $s$ (as a function of $m,n$ and $p$) such that from a random set of $mnp$ positive integers we can choose $snp$ numbers, such that they can be partitioned into $s$ sets of $np$ numbers, such that the sum of the numbers in every group gives the same remainder when divided by $p$.