Olympiad problems

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: primes , number theory , consecutive , divides , Sum of powers

Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.

2023 Belarus Team Selection Test, 1.3

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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2023 Belarus Team Selection Test, 2.1

Tags: number theory , divsibility , primes

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2022 AMC 10, 6

Tags: AMC , AMC 12 , AMC 10 , 2022 AMC 10B , 2022 AMC 12B , 2022 AMC , primes

How many of the first ten numbers of the sequence $121$, $11211$, $1112111$, ... are prime numbers? $\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$

2012 Mathcenter Contest + Longlist, 3

Tags: number theory , divisible , primes

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

2014 Czech-Polish-Slovak Junior Match, 3

Tags: number theory , primes , prime

Find with all integers $n$ when $|n^3 - 4n^2 + 3n - 35|$ and $|n^2 + 4n + 8|$ are prime numbers.

2024 SG Originals, 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]

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]

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]

2019 Cono Sur Olympiad, 4

Tags: number theory , primes , Diophantine equation , cono sur

Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

2013 Switzerland - Final Round, 2

Tags: inequalities , number theory , primes

Let $n$ be a natural number and $p_1, ..., p_n$ distinct prime numbers. Show that $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

2017 Hanoi Open Mathematics Competitions, 12

Tags: number theory , consecutive integers , primes

Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?

2022 Olimphíada, 1

Tags: Divisibility , primes

Let $p,q$ prime numbers such that $$p+q \mid p^3-q^3$$ Show that $p=q$.

1997 Israel National Olympiad, 3

Tags: inequalities , Product , primes , number theory

Let $n?$ denote the product of all primes smaller than $n$. Prove that $n? > n$ holds for any natural number $n > 3$.

2018 German National Olympiad, 5

Tags: number theory , Sequence , Prime factor , primes

We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence.

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 $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2025 Kosovo National Mathematical Olympiad`, P4

Tags: number theory , primes , GCD , functional equation , solve in natural , beautiful

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.

2015 India IMO Training Camp, 2

Tags: IMO Shortlist , number theory , primes , IMO Shortlist 2014 , N5

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2013 VJIMC, Problem 1

Tags: number theory , primes , prime

Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.

2019 AMC 10, 2

Tags: AMC , AMC 10 , AMC 10 B , 2019 AMC 10B , primes , AMC 12 B

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

1998 Estonia National Olympiad, 2

Tags: prime , primes , Digits , number theory

Find all prime numbers of the form $10101...01$.

2022 Cyprus JBMO TST, 2

Tags: number theory , primes , prime numbers

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

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.

1987 Polish MO Finals, 5

Tags: primes , Product , number theory , min

Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

2021 Nordic, 1

Tags: number theory , primes

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.