Olympiad problems

2019 IMEO, 4

Call a two-element subset of $\mathbb{N}$ [i]cute[/i] if it contains exactly one prime number and one composite number. Determine all polynomials $f \in \mathbb{Z}[x]$ such that for every [i]cute[/i] subset $ \{ p,q \}$, the subset $ \{ f(p) + q, f(q) + p \} $ is [i]cute[/i] as well. [i]Proposed by Valentio Iverson (Indonesia)[/i]

2015 Iran MO (2nd Round), 3

Tags: number theory , prime number , algebra

Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.

2001 National Olympiad First Round, 23

Tags: modular arithmetic , number theory , prime number

Which of the followings is false for the sequence $9,99,999,\dots$? $\textbf{(A)}$ The primes which do not divide any term of the sequence are finite. $\textbf{(B)}$ Infinitely many primes divide infinitely many terms of the sequence. $\textbf{(C)}$ For every positive integer $n$, there is a term which is divisible by at least $n$ distinct prime numbers. $\textbf{(D)}$ There is an inteter $n$ such that every prime number greater than $n$ divides infinitely many terms of the sequence. $\textbf{(E)}$ None of above

2009 Korea Junior Math Olympiad, 8

Tags: prime number , diophantine equation , number theory

Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

2024 Switzerland - Final Round, 1

Tags: number theory , prime number

If $a$ and $b$ are positive integers, we say that $a$ [i]almost divides[/i] $b$ if $a$ divides at least one of $b - 1$ and $b + 1$. We call a positive integer $n$ [i]almost prime[/i] if the following holds: for any positive integers $a, b$ such that $n$ almost divides $ab$, we have that $n$ almost divides at least one of $a$ and $b$. Determine all almost prime numbers. [hide = original link][url]https://mathematical.olympiad.ch/fileadmin/user_upload/Archiv/Intranet/Olympiads/Mathematics/deploy/exams/2024/FinalRound/Exam/englishFinalRound2024.pdf[/url]!![/hide]

2016 Uzbekistan National Olympiad, 2

Tags: number theory , prime number

$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.

2013 Korea Junior Math Olympiad, 4

Tags: number theory , prime number , minimum , divide

Prove that there exists a prime number $p$ such that the minimum positive integer $n$ such that $p|2^n -1$ is $3^{2013}$.

2011 IFYM, Sozopol, 5

Tags: number theory , prime number , sequence

Does there exist a strictly increasing sequence $\{a_n\}_{n=1}^\infty$ of natural numbers with the following property: for $\forall$ $c\in \mathbb{Z}$ the sequence $c+a_1,c+a_2,...,c+a_n...$ has finite number of primes? Explain your answer.

2013 Taiwan TST Round 1, 1

Tags: number theory , prime , prime number

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

2023 IMC, 4

Tags: algebra , polynomial , number theory , modular arithmetic , prime number

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_i=i^k+i$ for $i=0,1, \ldots,p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_2$ upon division by $p$?

2023 Poland - Second Round, 6

Tags: number theory , prime number , combinatorics , chessboard

Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

2012 Online Math Open Problems, 22

Tags: floor function , number theory , prime number

Find the largest prime number $p$ such that when $2012!$ is written in base $p$, it has at least $p$ trailing zeroes. [i]Author: Alex Zhu[/i]

2020 Kosovo National Mathematical Olympiad, 3

Tags: number theory , prime number , olym

Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.

2015 IFYM, Sozopol, 1

Tags: number theory , prime number , divisibility

Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , prime number

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

2019 BMT Spring, 6

Tags: number theory , prime number

Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.

2010 Iran Team Selection Test, 1

Tags: function , prime number , arithmetic sequence , number theory , logarithm

Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that \[f(a+1), f(a+2), \dots, f(a+n)\] form an arithmetic progression.

2006 JBMO ShortLists, 15

Tags: geometry , minimum , prime number , integer , area

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.

2008 Tuymaada Olympiad, 2

Tags: prime number , number theory

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2019 Korea Junior Math Olympiad., 5

Tags: number theory , prime number , algebra , polynomial

For prime number $p$, prove that there are integers $a$, $b$, $c$, $d$ such that for every integer $n$, the expression $n^4+1-\left( n^2+an+b \right) \left(n^2+cn+d \right)$ is a multiple of $p$.

2021 Turkey Team Selection Test, 1

Tags: legendre symbol , prime number , divisibility , number theory

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

1981 Czech and Slovak Olympiad III A, 4

Tags: number theory , combinatorics , prime number , sequence

Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.

2010 India IMO Training Camp, 8

Tags: prime number , number theory

Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

2004 China Western Mathematical Olympiad, 4

Tags: function , euler , search , number theory , totient function , prime number , logarithm

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2014 Bulgaria National Olympiad, 1

Tags: prime number , number theory

Find all pairs of prime numbers $p\,,q$ for which: \[p^2 \mid q^3 + 1 \,\,\, \text{and} \,\,\, q^2 \mid p^6-1\] [i]Proposed by P. Boyvalenkov[/i]