Olympiad problems

2010 Contests, 4

Determine the smallest positive integer $m$ with the property that $m^3-3m^2+2m$ is divisible by both $79$ and $83$.

2008 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: inequalities , abstract algebra , number theory , greatest common divisor , prime numbers

Prove that equation $ p^{4}\plus{}q^{4}\equal{}r^{4}$ does not have solution in set of prime numbers.

2009 All-Russian Olympiad, 6

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

Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?

2009 China Team Selection Test, 3

Tags: induction , number theory , prime numbers , relatively prime , prime factorization , greatest common divisor , Mobius function

Let $ (a_{n})_{n\ge 1}$ be a sequence of positive integers satisfying $ (a_{m},a_{n}) = a_{(m,n)}$ (for all $ m,n\in N^ +$). Prove that for any $ n\in N^ + ,\prod_{d|n}{a_{d}^{\mu (\frac {n}{d})}}$ is an integer. where $ d|n$ denotes $ d$ take all positive divisors of $ n.$ Function $ \mu (n)$ is defined as follows: if $ n$ can be divided by square of certain prime number, then $ \mu (1) = 1;\mu (n) = 0$; if $ n$ can be expressed as product of $ k$ different prime numbers, then $ \mu (n) = ( - 1)^k.$

2013 Tournament of Towns, 6

Tags: prime , Fraction , divides , divisor , number theory , prime numbers

The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

2005 Slovenia National Olympiad, Problem 2

Tags: number theory , prime numbers

Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.

1989 IMO Shortlist, 30

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 Iran MO (2nd Round), 1

Tags: number theory , prime numbers , MONT

a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$ b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$

2022 Bosnia and Herzegovina IMO TST, 2

Tags: number theory , prime numbers

Let $p$ be an odd prime number. Around a circular table, $p$ students sit. We give $p$ pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the $k-$th turn we skip $2k$ students and give the next student a piece of candy. We do this until we don't give out all $p$ pieces of candy. $a)$ How many students won't get any pieces of candy? $b)$ How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?

2016 Postal Coaching, 4

Tags: number theory , Diophantine equation , prime numbers

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

2016 IFYM, Sozopol, 7

Tags: number theory , prime numbers

Is the following set of prime numbers $p$ finite or infinite, where each $p$ [b]doesn't[/b] divide the numbers that can be expressed as $n^{2016}+2016^{2016}$ for $n\in \mathbb{N}$, if: a) $p=4k+3$; b) $p=4k+1$?

2016 Purple Comet Problems, 14

Tags: Purple Comet , number theory , prime numbers

Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

1994 IMO Shortlist, 3

Tags: modular arithmetic , number theory , prime numbers , Combinatorial Number Theory , IMO , IMO 1994 , combinatorics

Show that there exists a set $ A$ of positive integers with the following property: for any infinite set $ S$ of primes, there exist [i]two[/i] positive integers $ m$ in $ A$ and $ n$ not in $ A$, each of which is a product of $ k$ distinct elements of $ S$ for some $ k \geq 2$.

2010 Brazil National Olympiad, 2

Tags: algebra , polynomial , ceiling function , number theory , prime numbers , algebra unsolved

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2012 Singapore Junior Math Olympiad, 5

Tags: number theory , prime numbers , coprime , prime

Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number. (Note: Two positive integers $m, n$ are coprime if their only common factor is 1)

2005 CHKMO, 4

Tags: function , number theory , prime numbers , combinatorics unsolved , combinatorics

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

2007 South East Mathematical Olympiad, 3

Tags: ratio , number theory , prime numbers , number theory unsolved

Find all triples $(a,b,c)$ satisfying the following conditions: (i) $a,b,c$ are prime numbers, where $a<b<c<100$. (ii) $a+1,b+1,c+1$ form a geometric sequence.

2007 Italy TST, 3

Tags: inequalities , geometry , geometric transformation , reflection , number theory , prime numbers , number theory proposed

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

2020 Iran Team Selection Test, 6

Tags: number theory , prime numbers

$p$ is an odd prime number. Find all $\frac{p-1}2$-tuples $\left(x_1,x_2,\dots,x_{\frac{p-1}2}\right)\in \mathbb{Z}_p^{\frac{p-1}2}$ such that $$\sum_{i = 1}^{\frac{p-1}{2}} x_{i} \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{2} \equiv \cdots \equiv \sum_{i = 1}^{\frac{p-1}{2}} x_{i}^{\frac{p - 1}{2}} \pmod p.$$ [i]Proposed by Ali Partofard[/i]

2020 International Zhautykov Olympiad, 1

Tags: number theory , zhautykov , IZHO 2020 , Euler s Phi Function , Order , prime numbers

Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

2011 Czech-Polish-Slovak Match, 3

Tags: number theory , prime numbers , relatively prime , number theory unsolved

Let $a$ be any integer. Prove that there are infinitely many primes $p$ such that \[ p\,|\,n^2+3\qquad\text{and}\qquad p\,|\,m^3-a \] for some integers $n$ and $m$.

2023 Stars of Mathematics, 1

Tags: number theory , prime numbers , Perfect Squares

Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.

2023 Singapore Junior Math Olympiad, 4

Tags: number theory , prime numbers

Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.

2019 China Western Mathematical Olympiad, 7

Tags: number theory , prime numbers

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

2002 Korea Junior Math Olympiad, 2

Tags: number theory , prime numbers

Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.