Olympiad problems

1970 Polish MO Finals, 3

Prove that an integer $n > 1$ is a prime number if and only if, for every integer $k$ with $1\le k \le n-1$, the binomial coefficient $n \choose k$ is divisible by $n$.

2012 Dutch IMO TST, 1

Tags: perfect power , number theory , prime , gcd , greatest common divisor

For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2014 Contests, 4

Tags: number theory , point , circles , prime

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2008 Tournament Of Towns, 5

Tags: integer , combinatorics , prime

The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?

2016 Mediterranean Mathematics Olympiad, 4

Tags: number theory , prime

Determine all integers $n\ge1$ for which the number $n^8+n^6+n^4+4$ is prime. (Proposed by Gerhard Woeginger, Austria)

2012 APMO, 3

Tags: modular arithmetic , number theory , prime , divisibility

Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \frac{ n^p + 1 }{p^n + 1} $ is an integer.

1989 Austrian-Polish Competition, 6

Tags: sequence , prime , perfect square , number theory

A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

2012 Tournament of Towns, 5

Tags: prime , sum , divisible , number theory

Let $p$ be a prime number. A set of $p + 2$ positive integers, not necessarily distinct, is called [i]interesting [/i] if the sum of any $p$ of them is divisible by each of the other two. Determine all interesting sets.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime factorization , consecutive , prime

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

1965 Poland - Second Round, 4

Tags: number theory , prime

Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

2013 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , infinitely many , prime

Decide whether there are infinitely many primes $p$ having a multiple in the form $n^2 + n + 1$ for some natural number $n$

2013 Tournament of Towns, 6

Tags: prime , divide , divisor , number theory , prime number , fraction

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 $3n + 1$.

2003 Denmark MO - Mohr Contest, 3

Tags: number theory , prime

Determine the integers $n$ where $$|2n^2+9n+4|$$ is a prime number.

1995 IMO Shortlist, 6

Tags: modular arithmetic , number theory , counting , prime

Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

1987 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime , power , digit

Let $p>3$ be a prime and $n$ a positive integer such that $p^n$ has $20$ digits. Prove that at least one digit appears more than twice in this number.

2022 Cyprus JBMO TST, 2

Tags: number theory , prime number , prime

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

2010 Estonia Team Selection Test, 1

Tags: number theory , gcd , power of prime , coprime , prime

For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

1985 Spain Mathematical Olympiad, 4

Tags: number theory , positive integer , prime , sum of powers

Prove that for each positive integer $k $ there exists a triple $(a,b,c)$ of positive integers such that $abc = k(a+b+c)$. In all such cases prove that $a^3+b^3+c^3$ is not a prime.

2016 Korea Summer Program Practice Test, 3

Tags: number theory , prime number , prime , primitive root , decimal representation

Let $p > 10^9$ be a prime number such that $4p + 1$ is also prime. Prove that the decimal expansion of $\frac{1}{4p+1}$ contains all the digits $0,1, \ldots, 9$.

1979 IMO, 1

Tags: number theory , relatively prime , series summation , divisibility , prime

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

2009 Bosnia and Herzegovina Junior BMO TST, 3

Tags: number theory , prime , division , prime number

Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

2022 AMC 10, 13

Tags: prime , difference of cubes

The positive difference between a pair of primes is equal to $2$, and the positive difference between the cubes of the two primes is $31106$. What is the sum of the digits of the least prime that is greater than those two primes? $\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$

2017 Saudi Arabia BMO TST, 1

Tags: number theory , prime

Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers

2015 May Olympiad, 4

Tags: combinatorics , prime , number theory

The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.

2000 Mexico National Olympiad, 4

Tags: number theory , prime , integer sequence , maximum

Let $a$ and $b$ be positive integers not divisible by $5$. A sequence of integers is constructed as follows: the first term is $5$, and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$. (For example, if $a = 2$ and $b = 4$, the first three terms are $5,14,32$.) What is the maximum possible number of primes that can occur before encoutering the first composite term?