Olympiad problems

2009 Regional Olympiad of Mexico Center Zone, 2

Let $p \ge 2$ be a prime number and $a \ge 1$ a positive integer with $p \neq a$. Find all pairs $(a,p)$ such that: $a+p \mid a^2+p^2$

2001 Romania Team Selection Test, 2

Tags: function , pigeonhole principle , number theory , prime number , combinatorics , algebra

a) Let $f,g:\mathbb{Z}\rightarrow\mathbb{Z}$ be one to one maps. Show that the function $h:\mathbb{Z}\rightarrow\mathbb{Z}$ defined by $h(x)=f(x)g(x)$, for all $x\in\mathbb{Z}$, cannot be a surjective function. b) Let $f:\mathbb{Z}\rightarrow\mathbb{Z}$ be a surjective function. Show that there exist surjective functions $g,h:\mathbb{Z}\rightarrow\mathbb{Z}$ such that $f(x)=g(x)h(x)$, for all $x\in\mathbb{Z}$.

2016 Polish MO Finals, 1

Tags: polynomial , number theory , prime number , quadratic , algebra

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 $P_1, \ P_2 \in \mathbb{Z}[X]$.

2002 Olympic Revenge, 6

Tags: counting , number theory , prime number , combinatorics

Let $p$ a prime number, and $N$ the number of matrices $p \times p$ \[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1p}\\ a_{21} & a_{22} & \ldots & a_{2p}\\ \vdots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \ldots & a_{pp} \end{array}\] such that $a_{ij} \in \{0,1,2,\ldots,p\} $ and if $i \leq i^\prime$ and $j \leq j^\prime$, then $a_{ij} \leq a_{i^\prime j^\prime}$. Find $N \pmod{p}$.

2015 Caucasus Mathematical Olympiad, 1

Tags: number theory , prime , natural number , product , prime number

Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.

2022 USAMO, 4

Tags: number theory , diophantine equation , prime number

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

2020 OMpD, 3

Tags: algebra , polynomial , number theory , prime number

Determine all integers $n$ such that both of the numbers: $$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$ are both prime numbers.

2013 NIMO Problems, 10

Tags: number theory , prime number

There exist primes $p$ and $q$ such that \[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633 \times 134217730 \times 2^{2202} + 1. \] Find the remainder when $p+q$ is divided by $1000$. [i]Proposed by Evan Chen[/i]

2024 Kyiv City MO Round 1, Problem 2

Tags: number theory , prime number

Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $4 \times 4$ square, the sum of numbers in them is a prime number The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime. [img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

2018 PUMaC Live Round, 4.1

Tags: number theory , prime number

The number $400000001$ can be written as $p\cdot q$, where $p$ and $q$ are prime numbers. Find the sum of the prime factors of $p+q-1$.

2012 Online Math Open Problems, 22

Tags: floor function , number theory , prime number

Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

2011 China Western Mathematical Olympiad, 1

Tags: modular arithmetic , prime number , number theory

Does there exist any odd integer $n \geq 3$ and $n$ distinct prime numbers $p_1 , p_2, \cdots p_n$ such that all $p_i + p_{i+1} (i = 1,2,\cdots , n$ and $p_{n+1} = p_{1})$ are perfect squares?

2019 Macedonia Junior BMO TST, 1

Tags: number theory , prime number

Determine all prime numbers of the form $1 + 2^p + 3^p +...+ p^p$ where $p$ is a prime number.

2018 IFYM, Sozopol, 1

Tags: number theory , prime number

Find all prime numbers $p$ and all positive integers $n$, such that $n^8 - n^2 = p^5 + p^2$

2011 Greece Team Selection Test, 1

Tags: number theory , prime number

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

2005 IMO Shortlist, 3

Tags: algebra , polynomial , number theory , composite number , prime number

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

1969 AMC 12/AHSME, 23

Tags: factorial , number theory , prime number

For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

2010 Contests, 4

Tags: prime number , number theory

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2006 ISI B.Stat Entrance Exam, 3

Tags: induction , modular arithmetic , number theory , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2020 Peru IMO TST, 1

Tags: number theory , perfect power , prime number

Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2005 Iran MO (3rd Round), 5

Tags: induction , prime number , number theory

Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.

2020 Junior Balkan Team Selection Tests - Moldova, 10

Tags: number theory , prime number , perfect square

Find all pairs of prime numbers $(p, q)$ for which the numbers $p+q$ and $p+4q$ are simultaneously perfect squares.

2004 Iran MO (2nd round), 4

Tags: function , induction , prime number , number theory

$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that: \[f(m)+f(n) \ | \ m+n .\]

1955 Miklós Schweitzer, 4

Tags: number theory , prime number

[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]

LMT Speed Rounds, 2016.6

Tags: number theory , prime number

A positive integer is called [i]cool[/i] if it can be expressed in the form $a!\cdot b!+315$ where $a,b$ are positive integers. For example, $1!\cdot 1!+315=316$ is a cool number. Find the sum of all cool numbers that are also prime numbers. [i]Proposed by Evan Fang