Olympiad problems

2016 Iran Team Selection Test, 3

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 $p\nmid x_1x_2x_3x_4$.

2007 Balkan MO Shortlist, N5

Tags: prime numbers , Inequality , number theory

Let $p \geq 5$ be a prime and let \begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*} where $q_i$ are primes. Prove, \begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}

1996 National High School Mathematics League, 3

Tags: number theory , prime numbers

For a prime number $p$, there exists $n\in\mathbb{Z}_+$, $\sqrt{p+n}+\sqrt{n}$ is an integer, then $\text{(A)}$ there is no such $p$ $\text{(B)}$ there in only one such $p$ $\text{(C)}$ there is more than one such $p$, but finitely many $\text{(D)}$ there are infinitely many such $p$

2015 Azerbaijan JBMO TST, 4

Tags: number theory , prime numbers , AZE JBMO TST

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2018 Baltic Way, 16

Tags: number theory , prime numbers

Let $p$ be an odd prime. Find all positive integers $n$ for which $\sqrt{n^2-np}$ is a positive integer.

2011 IFYM, Sozopol, 6

Tags: number theory , prime numbers , modulo

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

1985 AMC 12/AHSME, 12

Tags: geometry , 3D geometry , number theory , prime numbers

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$

2015 Caucasus Mathematical Olympiad, 4

Tags: number theory , Sum , prime numbers , consecutive

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

1988 Brazil National Olympiad, 1

Tags: Brazilian Math Olympiad , Brazilian Math Olympiad 1988 , prime numbers , number theory

Find all primes which are sum of two primes and difference of two primes.

2022 Brazil Undergrad MO, 6

Tags: number theory , prime numbers

Let $p \equiv 3 \,(\textrm{mod}\, 4)$ be a prime and $\theta$ some angle such that $\tan(\theta)$ is rational. Prove that $\tan((p+1)\theta)$ is a rational number with numerator divisible by $p$, that is, $\tan((p+1)\theta) = \frac{u}{v}$ with $u, v \in \mathbb{Z}, v >0, \textrm{mdc}(u, v) = 1$ and $u \equiv 0 \,(\textrm{mod}\,p) $.

PEN C Problems, 4

Tags: floor function , quadratics , inequalities , number theory , prime numbers , Quadratic Residues

Let $M$ be an integer, and let $p$ be a prime with $p>25$. Show that the set $\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}$ contains a quadratic non-residue to modulus $p$.

2016 India IMO Training Camp, 3

Tags: number theory , set , prime numbers

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

1958 Poland - Second Round, 1

Tags: prime , number theory , prime numbers

Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.

1996 Bulgaria National Olympiad, 1

Tags: number theory , prime numbers

Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.

2006 ISI B.Stat Entrance Exam, 3

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

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

2019 Switzerland Team Selection Test, 4

Tags: number theory , polynomial , prime numbers , algebra

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

2019 India PRMO, 21

Tags: number theory , Sets , primes , prime numbers

Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition ${A, B}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number amonh these numbers. Find the sum of the digits of $N$.

the 11th XMO, 3

Tags: prime numbers , number theory

Let $p$ is a prime and $p\equiv 2\pmod 3$. For $\forall a\in\mathbb Z$, if $$p\mid \prod\limits_{i=1}^p(i^3-ai-1),$$then $a$ is called a "GuGu" number. How many "GuGu" numbers are there in the set $\{1,2,\cdots ,p\}?$ (We are allowed to discuss now. It is after 00:00 Feb 14 Beijing Time)

2000 Taiwan National Olympiad, 1

Tags: Euler , function , modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2004 Baltic Way, 10

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

Is there an infinite sequence of prime numbers $p_1$, $p_2$, $\ldots$, $p_n$, $p_{n+1}$, $\ldots$ such that $|p_{n+1}-2p_n|=1$ for each $n \in \mathbb{N}$?

2018 PUMaC Live Round, 4.1

Tags: PuMAC , Live Round , number theory , prime numbers

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 $p+q-1$.

2022 IMO Shortlist, N7

Tags: number theory , prime numbers , IMO , IMO 2022

Let $k$ be a positive integer and let $S$ be a finite set of odd prime numbers. Prove that there is at most one way (up to rotation and reflection) to place the elements of $S$ around the circle such that the product of any two neighbors is of the form $x^2+x+k$ for some positive integer $x$.

2009 Indonesia TST, 3

Tags: number theory , prime numbers , number theory proposed

Find integer $ n$ with $ 8001 < n < 8200$ such that $ 2^n \minus{} 1$ divides $ 2^{k(n \minus{} 1)! \plus{} k^n} \minus{} 1$ for all integers $ k > n$.

2020 Peru IMO TST, 1

Tags: Peru , TST , number theory , Perfect Powers , prime numbers

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 $k>1$

2025 Poland - First Round, 5

Tags: number theory , prime numbers

Positive integers $a, b, n$ are given. Assume that $a$ and $n$ are even, $b$ is odd and the number $ab(a+b)^{n-1}$ is divisible by $a^n+b^n$. Prove that there exist a prime number $p$, such that $p^{n+1}$ divides $a^n+b^n$.