Olympiad problems

2012 Tournament of Towns, 4

Let $C(n)$ be the number of prime divisors of a positive integer $n$. (a) Consider set $S$ of all pairs of positive integers $(a, b)$ such that $a \ne b$ and $C(a + b) = C(a) + C(b)$. Is $S$ finite or infinite? (b) Define $S'$ as a subset of S consisting of the pairs $(a, b)$ such that $C(a+b) > 1000$. Is $S'$ finite or infinite?

2019 Argentina National Olympiad, 5

Tags: number theory , arithmetic sequence , prime

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.

2015 IFYM, Sozopol, 5

Tags: number theory , divisibility , prime

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.

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 $p(x-y) = xy$.

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 $0,1, \ldots, 9$.

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$.

2012 Albania National Olympiad, 1

Tags: modular arithmetic , number theory , prime

Find all primes $p$ such that $p+2$ and $p^2+2p-8$ are also primes.

2020 Argentina National Olympiad, 4

Tags: prime , integer , number theory

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

2008 Tournament Of Towns, 5

Tags: combinatorics , prime , integer

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?

2024 Mexican Girls' Contest, 8

Tags: number theory , prime

Find all positive integers $n$ such that among the $n$ numbers \[ 2n + 1, \, 2^2 n + 1, \, \ldots, \, 2^n n + 1 \] there are $n$, $n - 1$, or $n - 2$ primes.

2019 Durer Math Competition Finals, 2

Tags: geometry , sidelengths , prime , number theory , right triangle , integer

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

2009 Bundeswettbewerb Mathematik, 2

Tags: number theory , quadratic trinomial , trinomial , prime , divisor , prime divisor

Let $n$ be an integer that is greater than $1$. Prove that the following two statements are equivalent: (A) There are positive integers $a, b$ and $c$ that are not greater than $n$ and for which that polynomial $ax^2 + bx + c$ has two different real roots $x_1$ and $x_2$ with $| x_2- x_1 | \le \frac{1}{n}$ (B) The number $n$ has at least two different prime divisors.

2000 Chile National Olympiad, 7

Tags: number theory , prime

Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.

2009 Postal Coaching, 2

Tags: prime , number theory , fraction

Find all pairs $(x, y)$ of natural numbers $x$ and $y$ such that $\frac{xy^2}{x+y}$ is a prime

2018 Saudi Arabia IMO TST, 1

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

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.

2000 Estonia National Olympiad, 1

Tags: number theory , power of prime , prime , remainder

Find all prime numbers whose sixth power does not give remainder $1$ when dividing by $504$

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 $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2012 Mathcenter Contest + Longlist, 2

Tags: number theory , prime

Let $p=2^n+1$ and $3^{(p-1)/2}+1\equiv 0 \pmod p$. Show that $p$ is a prime. [i](Zhuge Liang) [/i]

2005 Austrian-Polish Competition, 4

Tags: fermat's little theorem , prime , number theory , diophantine equation , perfect square

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

2023 Thailand TST, 1

Tags: number theory , prime , divisibility

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

1995 Korea National Olympiad, Day 2

Tags: polynomial , integer polynomial , greatest common divisor , prime , factorization , algebra

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: number theory , divide , prime , perfect square

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

2021 CHKMO, 2

Tags: number theory , prime factorization , prime

For each positive integer $n$ larger than $1$ with prime factorization $p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, its [i]signature[/i] is defined as the sum $\alpha_1+\alpha_2+\cdots+\alpha_k$. Does there exist $2020$ consecutive positive integers such that among them, there are exactly $1812$ integers whose signatures are strictly smaller than $11$?

2010 Belarus Team Selection Test, 2.2

Tags: number theory , prime , divisibility

Let $p$ be a positive prime integer, $S(p)$ be the number of triples $(x,y,z)$ such that $x,y,z\in\{0,1,..., p-1\}$ and $x^2+y^2+z^2$ is divided by $p$. Prove that $S(p) \ge 2p- 1$. (I. Bliznets)

2020 Nordic, 1

Tags: number theory , min , arithmetic progression , prime

For a positive integer $n$, denote by $g(n)$ the number of strictly ascending triples chosen from the set $\{1, 2, ..., n\}$. Find the least positive integer $n$ such that the following holds:[i] The number $g(n)$ can be written as the product of three different prime numbers which are (not necessarily consecutive) members in an arithmetic progression with common difference $336$.[/i]