Olympiad problems

2006 Thailand Mathematical Olympiad, 11

Let $p_n$ be the $n$-th prime number. Find the remainder when $\Pi_{n=1}^{2549} 2006^{p^2_{n-1}}$ is divided by $13$

2019 Centers of Excellency of Suceava, 2

Tags: prime , number theory , gcd

Let $ \left( s_n \right)_{n\ge 1 } $ be a sequence with $ s_1 $ and defined recursively as $ s_{n+1}=s_n^2-s_n+1. $ Prove that any two terms of this sequence are coprime. [i]Dan Nedeianu[/i]

2015 Chile National Olympiad, 2

Tags: prime , number theory

Find all prime numbers that do not have a multiple ending in $2015$.

2019 Finnish National High School Mathematics Comp, 2

Tags: floor function , prime , number theory , algebra

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

2012 Tournament of Towns, 2

Tags: prime , combinatorics , table , rectangle table

The cells of a $1\times 2n$ board are labelled $1,2,...,, n, -n,..., -2, -1$ from left to right. A marker is placed on an arbitrary cell. If the label of the cell is positive, the marker moves to the right a number of cells equal to the value of the label. If the label is negative, the marker moves to the left a number of cells equal to the absolute value of the label. Prove that if the marker can always visit all cells of the board, then $2n + 1$ is prime.

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.

2021 Puerto Rico Team Selection Test, 6

Tags: number theory , prime

Two positive integers $n,m\ge 2$ are called [i]allies[/i] if when written as a product of primes (not necessarily different): $n=p_1p_2...p_s$ and $m=q_1q_2...q_t$, turns out that: $$p_1 + p_2 + ... + p_s = q_1 + q_2 + ... + q_t$$ (a) Show that the biggest ally of any positive integer has to have only $2$ and $3$ in its prime factorization. (b) Find the biggest number which is allied of $2021$ .

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} $

2015 NIMO Summer Contest, 8

Tags: prime , number theory , exponent , large number

It is given that the number $4^{11}+1$ is divisible by some prime greater than $1000$. Determine this prime. [i] Proposed by David Altizio [/i]

2017 Puerto Rico Team Selection Test, 5

Tags: number theory , odd , prime

Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.

2015 Dutch Mathematical Olympiad, 4

Tags: number theory , prime , diophantine equation , diophantine

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

1978 Chisinau City MO, 159

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

Prove that the product of numbers $1, 2, ..., n$ ($n \ge 2$) is divisible by their sum if and only if the number $n + 1$ is not prime.

2023 Indonesia TST, 1

Tags: number theory , prime , divisibility

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

1996 Greece Junior Math Olympiad, 4b

Tags: diophantine equation , diophantine , prime , number theory

Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.

1999 Tournament Of Towns, 2

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

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)

2024 Abelkonkurransen Finale, 2a

Tags: sequence , algebra , prime

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 $a_n$.

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?

2008 Postal Coaching, 2

Tags: number theory , prime , prime number , divide

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

2013 Switzerland - Final Round, 2

Tags: inequalities , number theory , prime

Let $n$ be a natural number and $p_1, ..., p_n$ distinct prime numbers. Show that $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

2003 Olympic Revenge, 5

Tags: number theory , prime number , prime , divisibility

Let $[n]=\{1,2,...,n\}$.Let $p$ be any prime number. Find how many finite non-empty sets $S\in [p] \times [p]$ are such that $$\displaystyle \large p | \sum_{(x,y) \in S}{x},p | \sum_{(x,y) \in S}{y}$$

2004 Mexico National Olympiad, 1

Tags: perfect square , number theory , prime

Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

1997 Mexico National Olympiad, 1

Tags: number theory , prime number , prime

Determine all prime numbers $p$ for which $8p^4-3003$ is a positive prime number.

1979 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , divide , prime

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

2002 Croatia Team Selection Test, 3

Tags: number theory , prime

Prove that if $n$ is a natural number such that $1 + 2^n + 4^n$ is prime then $n = 3^k$ for some $k \in N_0$.

2009 Singapore Junior Math Olympiad, 3

Tags: digit , number theory , prime

Suppose $\overline{a_1a_2...a_{2009}}$ is a $2009$-digit integer such that for each $i = 1,2,...,2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$ (Note: $\overline{xyz...}$ denotes an integer whose digits are $x, y,z,...$.)