Olympiad problems

2022 Kazakhstan National Olympiad, 2

Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.

2022 IMO Shortlist, N6

Tags: number theory , prime factorization , function , prime numbers , primes

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

1999 Putnam, 6

Tags: Putnam , number theory , least common multiple , relatively prime , prime numbers , college contests , Putnam number theory

Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.

VMEO IV 2015, 12.3

Tags: number theory , arithmetic sequence , prime numbers

Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

1995 Czech and Slovak Match, 6

Tags: algebra , number theory , prime numbers

Find all triples $(x; y; p)$ of two non-negative integers $x, y$ and a prime number p such that $ p^x-y^p=1 $

2005 MOP Homework, 7

Tags: number theory , prime numbers , number theory unsolved

Let $A$ be a finite subset of prime numbers and $a> 1$ be a positive integer. Show that the number of positive integers $m$ for which all prime divisors of $a^m-1$ are in $A$ is finite.

1992 Turkey Team Selection Test, 1

Tags: number theory , prime numbers , number theory proposed

Is there $14$ consecutive positive integers such that each of these numbers is divisible by one of the prime numbers $p$ where $2\leq p \leq 11$.

2007 Tournament Of Towns, 5

Tags: number theory , prime numbers , number theory unsolved

Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.

2012 China Western Mathematical Olympiad, 1

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

Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$,have $105|9^{ p^2}-29^p+m.$ (September 28, 2012, Hohhot)

2011 CentroAmerican, 4

Tags: number theory , prime numbers , number theory proposed

Find all positive integers $p$, $q$, $r$ such that $p$ and $q$ are prime numbers and $\frac{1}{p+1}+\frac{1}{q+1}-\frac{1}{(p+1)(q+1)} = \frac{1}{r}.$

2004 Iran MO (2nd round), 4

Tags: function , induction , number theory , prime numbers , number theory proposed

$\mathbb{N}$ 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: \[f(m)+f(n) \ | \ m+n .\]

2003 IMO Shortlist, 8

Tags: modular arithmetic , number theory , prime numbers , Perfect Powers , IMO Shortlist

Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?

1996 IMO Shortlist, 1

Tags: number theory , prime numbers , game , invariant , IMO Shortlist

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

2016 Hong Kong TST, 4

Tags: number theory , prime numbers

Find all triples $(m,p,q)$ such that \begin{align*} 2^mp^2 +1=q^7, \end{align*} where $p$ and $q$ are ptimes and $m$ is a positive integer.

2002 Olympic Revenge, 6

Tags: counting , number theory , prime numbers , 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 $N \pmod{p}$.

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime numbers

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.

2021 Serbia Team Selection Test, P3

Tags: system of equations , number theory , Serbia , prime numbers , Diophantine Equations

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

2022 Cyprus JBMO TST, 2

Tags: number theory , primes , prime numbers

Determine all pairs of prime numbers $(p, q)$ which satisfy the equation \[ p^3+q^3+1=p^2q^2 \]

1997 Singapore Team Selection Test, 1

Tags: number theory , prime numbers , game , invariant , IMO Shortlist

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

2010 All-Russian Olympiad Regional Round, 9.8

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

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?

2023 India Regional Mathematical Olympiad, 2

Tags: number theory , prime numbers

Given a prime number $p$ such that $2p$ is equal to the sum of the squares of some four consecutive positive integers. Prove that $p-7$ is divisible by 36.

2006 Greece JBMO TST, 2

Tags: number theory , prime numbers , primes , exponential

Let $a,b,c$ be positive integers such that the numbers $k=b^c+a, l=a^b+c, m=c^a+b$ to be prime numbers. Prove that at least two of the numbers $k,l,m$ are equal.

2009 USAMTS Problems, 1

Tags: USAMTS , algorithm , inequalities , number theory , prime numbers

Archimedes planned to count all of the prime numbers between $2$ and $1000$ using the Sieve of Eratosthenes as follows: (a) List the integers from $2$ to $1000$. (b) Circle the smallest number in the list and call this $p$. (c) Cross out all multiples of $p$ in the list except for $p$ itself. (d) Let $p$ be the smallest number remaining that is neither circled nor crossed out. Circle $p$. (e) Repeat steps $(c)$ and $(d)$ until each number is either circled or crossed out. At the end of this process, the circled numbers are prime and the crossed out numbers are composite. Unfortunately, while crossing out the multiples of $2$, Archimedes accidentally crossed out two odd primes in addition to crossing out all the even numbers (besides $2$). Otherwise, he executed the algorithm correctly. If the number of circled numbers remaining when Archimedes finished equals the number of primes from $2$ to $1000$ (including $2$), then what is the largest possible prime that Archimedes accidentally crossed out?

2023 Brazil National Olympiad, 6

Tags: number theory , Sequence , prime numbers

For a positive integer $k$, let $p(k)$ be the smallest prime that does not divide $k$. Given a positive integer $a$, define the infinite sequence $a_0, a_1, \ldots$ by $a_0 = a$ and, for $n > 0$, $a_n$ is the smallest positive integer with the following properties: • $a_n$ has not yet appeared in the sequence, that is, $a_n \neq a_i$ for $0 \leq i < n$; • $(a_{n-1})^{a_n} - 1$ is a multiple of $p(a_{n-1})$. Prove that every positive integer appears as a term in the sequence, that is, for every positive integer $m$ there is $n$ such that $a_n = m$.

2003 Austrian-Polish Competition, 9

Tags: pigeonhole principle , number theory , prime numbers , combinatorics proposed , combinatorics

Take any 26 distinct numbers from {1, 2, ... , 100}. Show that there must be a non-empty subset of the $ 26$ whose product is a square. [hide] I think that the upper limit for such subset is 37.[/hide]