Olympiad problems

2017 Thailand TSTST, 2

Tags: Euler , function , modular arithmetic , quadratics , number theory , prime numbers , number theory unsolved

Suppose that for some $m,n\in\mathbb{N}$ we have $\varphi (5^m-1)=5^n-1$, where $\varphi$ denotes the Euler function. Show that $(m,n)>1$.

2019 Turkey Team SeIection Test, 2

Tags: number theory , Sequence , prime numbers

$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$. $a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite. $b)$ Find 3 different prime numbers that do not divide any terms of this sequence.

2005 Flanders Math Olympiad, 3

Tags: geometry , complex numbers , number theory , prime numbers , number theory proposed

Prove that $2005^2$ can be written in at least $4$ ways as the sum of 2 perfect (non-zero) squares.

2012-2013 SDML (Middle School), 10

Tags: number theory , prime numbers , SDML High School , 2012-13 SDML Round 2

Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

2022-23 IOQM India, 8

Tags: number theory , prime numbers , IOQM

Suppose the prime numbers $p$ and $q$ satisfy $q^2+3p=197p^2+q$.Write $\frac{p}{q}$ as $l+\frac{m}{n}$, where $l,m,n$ are positive integers , $m<n$ and $GCD(m,n)=1$. Find the maximum value of $l+m+n$.

2007 Thailand Mathematical Olympiad, 18

Tags: prime , remainder , number theory , prime numbers

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

1999 Baltic Way, 20

Tags: number theory , prime numbers , number theory proposed

Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .

2015 Portugal MO, 3

Tags: number theory , prime numbers

The numbers from $1$ to $2015$ are written on sheets so that if if $n-m$ is a prime, then $n$ and $m$ are on different sheets. What is the minimum number of sheets required?

2012 Belarus Team Selection Test, 1

Tags: number theory , prime numbers

Find all primes numbers $p$ such that $p^2-p-1$ is the cube of some integer.

2010 Contests, 2

Tags: algebra , polynomial , ceiling function , number theory , prime numbers , algebra unsolved

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2010 Morocco TST, 3

Tags: number theory , prime numbers

Any rational number admits a non-decimal representation unlimited decimal expansion. This development has the particularity of being periodic. Examples: $\frac{1}{7} = 0.142857142857…$ has a period $6$ while $\frac{1}{11}=0.0909090909 …$ $2$ periodic. What are the reciprocals of the prime integers with a period less than or equal to five?

2009 Jozsef Wildt International Math Competition, W. 5

Tags: number theory , prime numbers

Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.

2008 All-Russian Olympiad, 3

Tags: number theory , prime numbers , number theory unsolved

Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p \plus{} b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.

2021 Stars of Mathematics, 1

Tags: number theory , prime numbers , romania

For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$? [i]Russian Competition[/i]

1997 IMO Shortlist, 14

Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2024 Girls in Mathematics Tournament, 4

Tags: algebra , number theory , prime numbers

Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.

2023 Czech and Slovak Olympiad III A., 4

Tags: number theory , prime numbers

Let $(a_n)_{n = 0}^{\infty} $ be a sequence of positive integers such that for every $n \geq 0$ it is true that $$a_{n+2} = a_0 a_1 + a_1 a_2 + ... + a_n a_{n+1} - 1 $$ a) Prove that there exist a prime number which divides infinitely many $a_n$ b) Prove that there exist infinitely many such prime numbers

2006 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory , composite number , prime numbers , IMO Shortlist , Hi

Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$. Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.

2013 NIMO Problems, 10

Tags: number theory , prime numbers

There exist primes $p$ and $q$ such that \[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633 \times 134217730 \times 2^{2202} + 1. \] Find the remainder when $p+q$ is divided by $1000$. [i]Proposed by Evan Chen[/i]

2024 Baltic Way, 17

Tags: number theory proposed , prime numbers , Fermat s Little Theorem , Divisibility , Bound

Do there exist infinitely many quadruples $(a,b,c,d)$ of positive integers such that the number $a^{a!} + b^{b!} - c^{c!} - d^{d!}$ is prime and $2 \leq d \leq c \leq b \leq a \leq d^{2024}$?

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , prime numbers , 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 $0,1, \ldots, 9$.

2011 Puerto Rico Team Selection Test, 3

Tags: number theory , prime numbers , algebra unsolved , algebra

(a) Prove that (p^2)-1 is divisible by 24 if p is a prime number greater than 3. (b) Prove that (p^2)-(q^2) is divisible by 24 if p and q are prime numbers greater than 3.

2019 Pan-African, 2

Tags: number theory , prime numbers , Sums and Products , PAMO

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?

2020 Latvia TST, 1.5

Tags: combinatorics , prime numbers , combinatorics unsolved

Given a $6\times 6$ square consisting of unit squares, denote its rows and columns from $1$ to $6$. Figure [i]p-horse[/i] can move from square $(x; y)$ to $(x’; y’)$ if and only if both $x + x’$ and $y + y’$ are primes. At the start the [i]p-horse[/i] is located in one of the unit squares. $a)$ Can the [i]p-horse[/i] visit every unit square exactly once? $b$) Can the [i]p-horse[/i] visit every unit square exactly once and with the last move return to the initial starting position?

2019 Pan-African Shortlist, N2

Tags: number theory , prime numbers , Sums and Products , PAMO

Let $k$ be a positive integer. Consider $k$ not necessarily distinct prime numbers such that their product is ten times their sum. What are these primes and what is the value of $k$?