Olympiad problems

2010 Hanoi Open Mathematics Competitions, 7

Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers.

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 ($p$).

2017 International Zhautykov Olympiad, 2

Tags: number theory , divisor , prime

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2015 Belarus Team Selection Test, 1

Tags: number theory , divisible , consecutive , prime

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

1999 IMO Shortlist, 1

Tags: number theory , divisibility , prime

Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

2014 IMO Shortlist, N5

Tags: number theory , prime

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]

1995 IMO, 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$?

2012 Dutch IMO TST, 1

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

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 Junior Balkan Team Selection Tests - Romania, 3

Tags: sum , combinatorics , number theory , prime

Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

1997 Israel National Olympiad, 3

Tags: inequalities , product , number theory , prime

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

2016 Lusophon Mathematical Olympiad, 5

Tags: number theory , prime number , prime , sequence

A numerical sequence is called lusophone if it satisfies the following three conditions: i) The first term of the sequence is number $1$. ii) To obtain the next term of the sequence we can multiply the previous term by a positive prime number ($2,3,5,7,11, ...$) or add $1$. (iii) The last term of the sequence is the number $2016$. For example: $1\overset{{\times 11}}{\to}11 \overset{{\times 61}}{\to} 671 \overset{{+1}}{\to}672 \overset{{\times 3}}{\to}2016$ How many Lusophone sequences exist in which (as in the example above) the add $1$ operation was used exactly once and not multiplied twice by the same prime number?

2015 Taiwan TST Round 2, 3

Tags: number theory , prime

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]

1998 Estonia National Olympiad, 4

Tags: divide , number theory , divisible , prime

Prove that if for a positive integer $n$ is $5^n + 3^n + 1$ is prime number, then $n$ is divided by $12$.

2009 Tournament Of Towns, 7

Tags: gcd , number theory , prime

Initially a number $6$ is written on a blackboard. At $n$-th step an integer $k$ on the blackboard is replaced by $k+gcd(k,n)$. Prove that at each step the number on the blackboard increases either by $1$ or by a prime number.

2020 Dutch BxMO TST, 5

Tags: number theory , product , prime

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

2025 Kosovo National Mathematical Olympiad`, P4

Tags: number theory , gcd , functional equation , solve in natural , prime

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 $f(p^{2025})=q^{2025}$.

2011 Silk Road, 4

Tags: number theory , representation , prime

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

1994 Chile National Olympiad, 3

Tags: prime , number theory , digit

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

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 $r$.

2001 Estonia Team Selection Test, 5

Tags: product , power , digit , prime , number theory

Find the exponent of $37$ in the representation of the number $111...... 11$ with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers

2014 India PRMO, 1

Tags: number theory , prime , factor , maximum

A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

2022 AMC 10, 13

Tags: difference of cubes , prime

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? $\textbf{(A) } 8 \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 11 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 16$