Olympiad problems

2014 India PRMO, 1

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

2012 Brazil Team Selection Test, 3

Tags: modular arithmetic , number theory , prime , divisibility

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

2008 Postal Coaching, 2

Tags: number theory , prime , prime number , divide

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2015 Chile National Olympiad, 2

Tags: number theory , prime

Find all prime numbers that do not have a multiple ending in $2015$.

1987 Polish MO Finals, 5

Tags: product , number theory , min , prime

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

2013 Thailand Mathematical Olympiad, 1

Tags: divide , number theory , prime

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , prime , remainder

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

2000 Tuymaada Olympiad, 5

Tags: number theory , divisible , prime

Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

2010 Belarus Team Selection Test, 2.2

Tags: number theory , prime , divisibility

Let $p$ be a positive prime integer, $S(p)$ be the number of triples $(x,y,z)$ such that $x,y,z\in\{0,1,..., p-1\}$ and $x^2+y^2+z^2$ is divided by $p$. Prove that $S(p) \ge 2p- 1$. (I. Bliznets)

2025 International Zhautykov Olympiad, 3

Tags: number theory , construction , prime

A pair of positive integers $(x, y)$ is [i] good [/i] if they satisfy $\text{rad}(x) = \text{rad}(y)$ and they do not divide each-other. Given coprime positive integers $a$ and $b$, show that there exist infinitely many $n$ for which there exists a positive integer $m$ such that $(a^n + bm, b^n + am)$ is [i] good[/i]. (Here, $\text{rad}(x)$ denotes the product of $x$'s prime divisors, as usual.)

2019 Peru EGMO TST, 1

Tags: diophantine equation , diophantine , number theory , prime

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

2015 NIMO Summer Contest, 11

Tags: square , prime

We say positive integer $n$ is $\emph{metallic}$ if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers? [i] Proposed by Lewis Chen [/i]

1962 Czech and Slovak Olympiad III A, 1

Tags: perfect square , prime , algebra , quadratic polynomial

Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.

2006 Greece JBMO TST, 2

Tags: number theory , prime number , prime , exponential

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

2022 Indonesia TST, N

Tags: polynomial , number theory , prime , product , infinity

Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

2015 Indonesia MO Shortlist, N1

Tags: number theory , perfect square , triplets , prime

A triple integer $(a, b, c)$ is called [i]brilliant [/i] when it satisfies: (i) $a> b> c$ are prime numbers (ii) $a = b + 2c$ (iii) $a + b + c$ is a perfect square number Find the minimum value of $abc$ if triple $(a, b, c)$ is [i]brilliant[/i].

2010 Regional Olympiad of Mexico Center Zone, 2

Tags: perfect power , number theory , prime

Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

Mathematical Minds 2024, P6

Tags: recursion , number theory , sequence , prime

Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

2022 AMC 10, 6

Tags: prime

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

2015 NIMO Summer Contest, 8

Tags: prime , number theory , exponent , large number

It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime. [i] Proposed by David Altizio [/i]

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.

2022 239 Open Mathematical Olympiad, 5

Tags: number theory , prime

Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

2000 Singapore MO Open, 2

Tags: divide , divisible , gcd , prime , number theory

Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

2015 Belarus Team Selection Test, 1

Tags: number theory , prime

Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$. B.Gilevich

2025 Bulgarian Winter Tournament, 11.4

Tags: sequence , divisibility , infinite prime divisors , function , polynomial , prime , algebra

Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.