Olympiad problems

2009 Germany Team Selection Test, 2

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

2010 All-Russian Olympiad Regional Round, 9.8

Tags: quadratic , prime number , number theory

For every positive integer $n$, let $S_n$ be the sum of the first $n$ prime numbers: $S_1 = 2, S_2 = 2 + 3 = 5, S_3 = 2 + 3 + 5 = 10$, etc. Can both $S_n$ and $S_{n+1}$ be perfect squares?

2018 Malaysia National Olympiad, A6

Tags: number theory , prime number , power

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

2022 Korea Junior Math Olympiad, 3

Tags: number theory , prime number

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: [list] [*]$a_1=1$ [*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. [/list] Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.

2023 Kazakhstan National Olympiad, 5

Tags: number theory , prime number

Solve the given equation in prime numbers $$p^3+q^3+r^3=p^2qr$$

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime number

Let $n$ and $k$ be two positive integers such that $1\leq n \leq k$. Prove that, if $d^k+k$ is a prime number for each positive divisor $d$ of $n$, then $n+k$ is a prime number.

2015 Turkey Junior National Olympiad, 3

Tags: number theory , diophantine equation , prime number

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

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 $p$.

2022 Assara - South Russian Girl's MO, 5

Tags: number theory , prime number , prime

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

2013 Korea - Final Round, 5

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

Two coprime positive integers $ a, b $ are given. Integer sequence $ \{ a_n \}, \{b_n \} $ satisties \[ (a+b \sqrt2 )^{2n} = a_n + b_n \sqrt2 \] Find all prime numbers $ p $ such that there exist positive integer $ n \le p $ satisfying $ p | b_n $.

2019 Pan-African, 2

Tags: number theory , prime number , sums and products

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

2010 Contests, 1

Tags: prime number , number theory

$a)$ Let $p$ and $q$ be distinct prime numbers such that $p+q^2$ divides $p^2+q$. Prove that $p+q^2$ divides $pq-1$. $b)$ Find all prime numbers $p$ such that $p+121$ divides $p^2+11$.

2021 Turkey MO (2nd round), 5

Tags: number theory , prime number

There are finitely many primes dividing the numbers $\{ a \cdot b^n + c\cdot d^n : n=1, 2, 3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.

2005 Flanders Junior Olympiad, 3

Tags: geometry , complex number , prime number , number theory

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2017 Iran Team Selection Test, 4

Tags: number theory , prime number , modular arithmetic

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

1996 Iran MO (3rd Round), 4

Tags: prime number , combinatorics , number theory

Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that \[\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.\]

1977 IMO Longlists, 27

Tags: number theory , prime number , prime factorization , dirichlet s theorem

Let $n$ be a given number greater than 2. We consider the set $V_n$ of all the integers of the form $1 + kn$ with $k = 1, 2, \ldots$ A number $m$ from $V_n$ is called indecomposable in $V_n$ if there are not two numbers $p$ and $q$ from $V_n$ so that $m = pq.$ Prove that there exist a number $r \in V_n$ that can be expressed as the product of elements indecomposable in $V_n$ in more than one way. (Expressions which differ only in order of the elements of $V_n$ will be considered the same.)

2007 ITest, 1

Tags: number theory , prime number

A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.) $\textbf{(A) }4$

2015 Belarus Team Selection Test, 3

Tags: combinatorics , algebra , prime number

Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

1988 IMO Longlists, 94

Tags: number theory , prime number , perfect square

Let $n+1, n \geq 1$ positive integers be formed by taking the product of $n$ given prime numbers (a prime number can appear several times or also not appear at all in a product formed in this way.) Prove that among these $n+1$ one can find some numbers whose product is a perfect square.

2011 IFYM, Sozopol, 6

Tags: number theory , prime number , modulo

Find all prime numbers $p$ for which $x^4\equiv -1\, (mod\, p)$ has a solution.

2023 Olimphíada, 4

Tags: floor function , prime number , golden ratio , residue , number theory

We say that a prime $p$ is $n$-$\textit{rephinado}$ if $n | p - 1$ and all $1, 2, \ldots , \lfloor \sqrt[\delta]{p}\rfloor$ are $n$-th residuals modulo $p$, where $\delta = \varphi+1$. Are there infinitely many $n$ for which there are infinitely many $n$-$\textit{rephinado}$ primes? Notes: $\varphi =\frac{1+\sqrt{5}}{2}$. We say that an integer $a$ is a $n$-th residue modulo $p$ if there is an integer $x$ such that $$x^n \equiv a \text{ (mod } p\text{)}.$$

1992 Spain Mathematical Olympiad, 4

Tags: number theory , prime number , arithmetic sequence

Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.

2022 Bulgarian Spring Math Competition, Problem 12.4

Tags: number theory , prime number , 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 $p$.

2016 Iran Team Selection Test, 3

Tags: number theory , prime number , modular arithmetic , quadratic residue

Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$ has a solution in integers such that $p\nmid x_1x_2x_3x_4$.