Olympiad problems

2004 Finnish National High School Mathematics Competition, 4

The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number. Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?

2009 Germany Team Selection Test, 1

Tags: prime number , number theory

For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$

2018 Mediterranean Mathematics OIympiad, 3

Tags: number theory , prime number

An integer $a\ge1$ is called [i]Aegean[/i], if none of the numbers $a^{n+2}+3a^n+1$ with $n\ge1$ is prime. Prove that there are at least 500 Aegean integers in the set $\{1,2,\ldots,2018\}$. (Proposed by Gerhard Woeginger, Austria)

2010 Albania Team Selection Test, 4

Tags: algebra , polynomial , function , inequalities , prime number , number theory

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

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.

1994 APMO, 3

Tags: relatively prime , prime number , number theory

Let $n$ be an integer of the form $a^2 + b^2$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \leq \sqrt{n}$, then $p$ divides $ab$. Determine all such $n$.

1993 All-Russian Olympiad, 1

Tags: geometry , perimeter , inequalities , number theory , prime number , area of a triangle , heron's formula

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

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.

2019 Dutch IMO TST, 4

Tags: number theory , function , find all functions , prime number , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2013 Peru IMO TST, 5

Tags: number theory , prime number , binomial coefficient

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2020-IMOC, N3

Tags: number theory , prime number

$\textbf{N3:}$ For any positive integer $n$, define $rad(n)$ to be the product of all prime divisors of $n$ (without multiplicities), and in particular $rad(1)=1$. Consider an infinite sequence of positive integers $\{a_n\}_{n=1}^{\infty}$ satisfying that \begin{align*} a_{n+1} = a_n + rad(a_n), \: \forall n \in \mathbb{N} \end{align*} Show that there exist positive integers $t,s$ such that $a_t$ is the product of the $s$ smallest primes. [i]Proposed by ltf0501[/i]

2016 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime number , infinitely many solutions

Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.

2009 Regional Olympiad of Mexico Center Zone, 2

Tags: number theory , prime number

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: $a+p \mid a^2+p^2$

2021 Kyiv City MO Round 1, 11.5

Tags: number theory , prime number

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

2018 IFYM, Sozopol, 5

Tags: number theory , prime number

Find the solutions in prime numbers of the following equation $p^4 + q^4 + r^4 + 119 = s^2 .$

2012 Morocco TST, 1

Tags: number theory , prime number

Find all prime numbers $p_1,…,p_n$ (not necessarily different) such that : $$ \prod_{i=1}^n p_i=10 \sum_{i=1}^n p_i$$

2018 Baltic Way, 20

Tags: number theory , prime number

Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.

2018 AMC 10, 5

Tags: number theory , prime number

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

1985 AMC 12/AHSME, 12

Tags: geometry , 3d geometry , number theory , prime number

Let's write p,q, and r as three distinct prime numbers, where 1 is not a prime. Which of the following is the smallest positive perfect cube leaving $ n \equal{} pq^2r^4$ as a divisor? $ \textbf{(A)}\ p^8q^8r^8\qquad \textbf{(B)}\ (pq^2r^2)^3\qquad \textbf{(C)}\ (p^2q^2r^2)^3\qquad \textbf{(D)}\ (pqr^2)^3\qquad \textbf{(E)}\ 4p^3q^3r^3$

2014 Ukraine Team Selection Test, 9

Tags: number theory , prime number , parameterization , diophantine , system of equations , diophantine equation

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

2015 JBMO Shortlist, NT4

Tags: number theory , prime number

Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] Proposed by Moldova

2022 IOQM India, 3

Tags: number theory , prime number , arithmetic sequence , perimeter , trivial , geometry

Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$.

1998 AMC 12/AHSME, 12

Tags: number theory , prime number , logarithm

How many different prime numbers are factors of $ N$ if \[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11? \]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$

1975 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime number

Prove that no term of the sequence $10001$, $100010001$, $1000100010001$ , $...$ is prime.

2016 Lusophon Mathematical Olympiad, 5

Tags: number theory , prime number , prime , sequence

A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number $1$. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$. (iii) The last term of the sequence is the number $2016$. For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$ How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?