Olympiad problems

2006 Estonia Team Selection Test, 6

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

2018 Polish Junior MO First Round, 3

Tags: number theory , prime numbers , Divisibility , Poland

Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.

1999 China Team Selection Test, 2

Tags: number theory , prime numbers , number theory unsolved

Find all prime numbers $p$ which satisfy the following condition: For any prime $q < p$, if $p = kq + r, 0 \leq r < q$, there does not exist an integer $q > 1$ such that $a^{2} \mid r$.

2024 Thailand TSTST, 4

Tags: number theory , prime numbers

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

2020-21 KVS IOQM India, 11

Tags: IOQM , KV , number theory , prime numbers

The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2$. Determine the sum of all possible values of $a+b+c$.

2015 JBMO TST - Turkey, 1

Tags: number theory , prime numbers

Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy: $$p+q=r(p-q)^n$$ [i]Proposed by Şahin Emrah[/i]

2016 JBMO TST - Turkey, 7

Tags: number theory , prime numbers , DUMB BOUNDING

Find all pairs $(p, q)$ of prime numbers satisfying \[ p^3+7q=q^9+5p^2+18p. \]

1989 IMO Longlists, 93

Tags: modular arithmetic , number theory , prime numbers , Sequence , power of number , IMO , IMO 1989

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2017 Israel Oral Olympiad, 3

Tags: number theory , prime , prime numbers

2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2023 Iran MO (3rd Round), 2

Tags: number theory , prime numbers

Let $N$ be the number of ordered pairs $(x,y)$ st $1 \leq x,y \leq p(p-1)$ and : $$x^{y} \equiv y^{x} \equiv 1 \pmod{p}$$ where $p$ is a fixed prime number. Show that : $$(\phi {(p-1)}d(p-1))^2 \leq N \leq ((p-1)d(p-1))^2$$ where $d(n)$ is the number of divisors of $n$

2008 Postal Coaching, 2

Tags: number theory , primes , prime , prime numbers , divides

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]

1974 IMO Longlists, 2

Tags: number theory , prime numbers , number theory proposed

Let ${u_n}$ be the Fibonacci sequence, i.e., $u_0=0,u_1=1,u_n=u_{n-1}+u_{n-2}$ for $n>1$. Prove that there exist infinitely many prime numbers $p$ that divide $u_{p-1}$.

2015 Romanian Master of Mathematics, 5

Tags: RMM , number theory , prime numbers

Let $p \ge 5$ be a prime number. For a positive integer $k$, let $R(k)$ be the remainder when $k$ is divided by $p$, with $0 \le R(k) \le p-1$. Determine all positive integers $a < p$ such that, for every $m = 1, 2, \cdots, p-1$, $$ m + R(ma) > a. $$

2024 Singapore MO Open, Q5

Tags: combinatorics , primes , grid , number theory , prime numbers

Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2007 All-Russian Olympiad Regional Round, 8.3

Tags: number theory , prime numbers , number theory proposed

Determine if there exist prime numbers $ p_{1},p_{2},...,p_{2007}$ such that $ p_{2}|p_{1}^{2}\minus{}1,p_{3}|p_{2}^{2}\minus{}1,...,p_{1}|p_{2007}^{2}\minus{}1$.

2021 IMC, 6

Tags: IMC 2021 , group theory , number theory , prime numbers , abstract algebra

For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.

2013 IFYM, Sozopol, 3

Tags: number theory , prime numbers

Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$

2006 JBMO ShortLists, 15

Tags: geometry , areas , minimum , prime numbers , Integer

Let $A_1$ and $B_1$ be internal points lying on the sides $BC$ and $AC$ of the triangle $ABC$ respectively and segments $AA_1$ and $BB_1$ meet at $O$. The areas of the triangles $AOB_1,AOB$ and $BOA_1$ are distinct prime numbers and the area of the quadrilateral $A_1OB_1C$ is an integer. Find the least possible value of the area of the triangle $ABC$, and argue the existence of such a triangle.

2010 Poland - Second Round, 3

Tags: inequalities , number theory , prime numbers , number theory proposed

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

2017 CentroAmerican, 3

Tags: OMCC , OMCC 2017 , CENTRO , number theory , prime numbers

Tita the Frog sits on the number line. She is initially on the integer number $k>1$. If she is sitting on the number $n$, she hops to the number $f(n)+g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct integers.

1999 AMC 12/AHSME, 4

Tags: number theory , prime numbers

Find the sum of all prime numbers between $ 1$ and $ 100$ that are simultaneously $ 1$ greater than a multiple of $ 4$ and $ 1$ less than a multiple of $ 5$. $ \textbf{(A)}\ 118\qquad \textbf{(B)}\ 137\qquad \textbf{(C)}\ 158\qquad \textbf{(D)}\ 187 \qquad \textbf{(E)}\ 245$

2024 Francophone Mathematical Olympiad, 4

Tags: number theory , number theory proposed , prime numbers , Divisors

Let $p$ be a fixed prime number. Find all integers $n \ge 1$ with the following property: One can partition the positive divisors of $n$ in pairs $(d,d')$ satisfying $d<d'$ and $p \mid \left\lfloor \frac{d'}{d}\right\rfloor$.

2010 China Team Selection Test, 2

Tags: number theory , prime numbers , factorial

Prove that there exists a sequence of unbounded positive integers $a_1\leq a_2\leq a_3\leq\cdots$, such that there exists a positive integer $M$ with the following property: for any integer $n\geq M$, if $n+1$ is not prime, then any prime divisor of $n!+1$ is greater than $n+a_n$.

2023 Olimphíada, 4

Tags: floor function , prime numbers , 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 $$x^n \equiv a \text{ (mod } p\text{)}.$$

2009 Regional Olympiad of Mexico Center Zone, 2

Tags: number theory , prime numbers

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