Olympiad problems

2002 IMO Shortlist, 3

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2022 Assara - South Russian Girl's MO, 5

Tags: number theory , prime number , prime

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

2012 NZMOC Camp Selection Problems, 4

Tags: number theory , multiple , divide , divisible , prime

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2018 Saudi Arabia IMO TST, 1

Tags: increasing , sequence , number theory , prime , algebra

Consider the infinite, strictly increasing sequence of positive integer $(a_n)$ such that i. All terms of sequences are pairwise coprime. ii. The sum $\frac{1}{\sqrt{a_1a_2}} +\frac{1}{\sqrt{a_2a_3}}+ \frac{1}{\sqrt{a_3a_4}} + ..$ is unbounded. Prove that this sequence contains infinitely many primes.

2016 Croatia Team Selection Test, Problem 4

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 $0,1, \ldots, 9$.

2021 South Africa National Olympiad, 3

Tags: perfect square , number theory , sum of squares , power of 2 , prime

Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.

1996 Estonia National Olympiad, 1

Tags: prime , diophantine equation , diophantine , number theory

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

1994 Abels Math Contest (Norwegian MO), 2a

Tags: diophantine , number theory , diophantine equation , prime

Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2014 Finnish National High School Mathematics, 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 $r$.

2007 Korea Junior Math Olympiad, 8

Tags: prime , positive integer , divisor , number theory

Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infinite number of [i]Primes of the Year[/i].

2019 Bundeswettbewerb Mathematik, 4

Tags: number theory , prime

Prove that for no integer $k \ge 2$, between $10k$ and $10k + 100$ there are more than $23$ prime numbers.

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divide

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2008 May Olympiad, 3

Tags: number theory , prime

In numbers $1010... 101$ Ones and zeros alternate, if there are $n$ ones, there are $n -1$ zeros ($n \ge 2$ ).Determine the values of $n$ for which the number $1010... 101$, which has $n$ ones, is prime.

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 $gcd(n, n @m) = 1$.

2015 Ukraine Team Selection Test, 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]

1997 Czech And Slovak Olympiad IIIA, 4

Tags: sequence , prime , number theory

Show that there exists an increasing sequence $a_1,a_2,a_3,...$ of natural numbers such that, for any integer $k \ge 2$, the sequence $k+a_n$ ($n \in N$) contains only finitely many primes.

2003 Junior Tuymaada Olympiad, 2

Tags: number theory , prime , perfect square

Find all natural $ x $ for which $ 3x+1 $ and $ 6x-2 $ are perfect squares, and the number $ 6x^2-1 $ is prime.

1989 Austrian-Polish Competition, 3

Tags: number theory , digit , prime

Find all natural numbers $N$ (in decimal system) with the following properties: (i) $N =\overline{aabb}$, where $\overline{aab}$ and $\overline{abb}$ are primes, (ii) $N = P_1P_2P_3$, where $P_k (k = 1,2,3)$ is a prime consisting of $k$ (decimal) digits.

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.

2014 Dutch Mathematical Olympiad, 4

Tags: number theory , prime , divisible , inequality , diophantine

A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

1978 Bundeswettbewerb Mathematik, 4

Tags: prime , decimal representation , permutation , number theory

A prime number has the property that however its decimal digits are permuted, the obtained number is also prime. Prove that this number has at most three different digits. Also prove a stronger statement.

2013 Ukraine Team Selection Test, 11

Tags: number theory , divisibility , prime

Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.

2014 Indonesia MO Shortlist, N1

Tags: number theory , diophantine equation , prime

(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number. (b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.

1984 Tournament Of Towns, (060) A5

Tags: number theory , prime , consecutive

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)

1993 Mexico National Olympiad, 6

Tags: number theory , prime , divisibility

$p$ is an odd prime. Show that $p$ divides $n(n+1)(n+2)(n+3) + 1$ for some integer $n$ iff $p$ divides $m^2 - 5$ for some integer $m$.