Olympiad problems

2015 India IMO Training Camp, 2

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]

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 $p(x-y) = xy$.

2024 Abelkonkurransen Finale, 2a

Tags: prime , sequence , algebra

Positive integers $a_0<a_1<\dots<a_n$, are to be chosen so that $a_j-a_i$ is not a prime for any $i,j$ with $0 \le i <j \le n$. For each $n \ge 1$, determine the smallest possible value of $a_n$.

2009 Bundeswettbewerb Mathematik, 2

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

2023 Switzerland - Final Round, 6

Tags: number theory , prime , divisibility

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

2018 Costa Rica - Final Round, N4

Tags: prime , number theory

Let $p$ be a prime number such that $p = 10^{d -1} + 10^{d-2} + ...+ 10 + 1$. Show that $d$ is a prime.

1949-56 Chisinau City MO, 7

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

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2012 Dutch IMO TST, 1

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

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

2016 Croatia Team Selection Test, Problem 4

Tags: prime , diophantine equation , prime number , number theory

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2022 Indonesia TST, N

Tags: number theory , infinity , product , prime , polynomial

Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

2003 Denmark MO - Mohr Contest, 3

Tags: prime , number theory

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

2008 May Olympiad, 3

Tags: prime , number theory

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.

1993 ITAMO, 2

Tags: quadratic trinomial , rational roots , rational , prime , quadratic , trinomial , number theory

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

2016 Croatia Team Selection Test, Problem 4

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

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 $0,1, \ldots, 9$.

1997 Austrian-Polish Competition, 5

Tags: polynomial , prime , integer polynomial , cubic

Let $p_1,p_2,p_3,p_4$ be four distinct primes. Prove that there is no polynomial $Q(x) = ax^3 + bx^2 + cx + d$ with integer coefficients such that $|Q(p_1)| =|Q(p_2)| = |Q(p_3)|= |Q(p_4 )| = 3$.

2021 Nordic, 1

Tags: prime , number theory

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

2024 Mexican Girls' Contest, 8

Tags: prime , number theory

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.

2013 VJIMC, Problem 1

Tags: prime , number theory

Let $S_n$ denote the sum of the first $n$ prime numbers. Prove that for any $n$ there exists the square of an integer between $S_n$ and $S_{n+1}$.

2013 Taiwan TST Round 1, 1

Tags: number theory , prime , prime number

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

2012 Mathcenter Contest + Longlist, 3

Tags: prime , divisible , number theory

If $p,p^2+2$ are both primes, how many divisors does $p^5+2p^2$ have? [i](Zhuge Liang)[/i]

2019 Cono Sur Olympiad, 4

Tags: number theory , prime , diophantine equation

Find all positive prime numbers $p,q,r,s$ so that $p^2+2019=26(q^2+r^2+s^2)$.

1999 Tournament Of Towns, 2

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

Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$. (a) May it happen that $d = 2$? (b) May it happen that $d$ is prime? (V Senderov)

1997 Slovenia Team Selection Test, 6

Tags: prime , number theory

Let $p$ be a prime number and $a$ be an integer. Prove that if $2^p +3^p = a^n$ for some integer $n$, then $n = 1$.

2018 Federal Competition For Advanced Students, P2, 3

Tags: prime , combinatorics , number theory

There are $n$ children in a room. Each child has at least one piece of candy. In Round $1$, Round $2$, etc., additional pieces of candy are distributed among the children according to the following rule: In Round $k$, each child whose number of pieces of candy is relatively prime to $k$ receives an additional piece. Show that after a sufficient number of rounds the children in the room have at most two different numbers of pieces of candy. [i](Proposed by Theresia Eisenkölbl)[/i]

2001 Estonia Team Selection Test, 5

Tags: prime , product , power , digit , 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