Olympiad problems

2011 Indonesia TST, 4

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 $2^p - 1|N$.

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.

2018 Chile National Olympiad, 1

Tags: prime number , prime , number theory

Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

2011 Silk Road, 4

Tags: prime , representation , number theory

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .

2013 QEDMO 13th or 12th, 6

Tags: prime , number theory

A composite natural number $n$ is called [i]happy [/i] if at most one of the numbers $2^{2^n}+ 1$ and $6^{2^n}+ 1$ is prime. Show that there are infinitely many happy numbers.

1993 Spain Mathematical Olympiad, 4

Tags: prime , multiple , number theory

Prove that for each prime number distinct from $2$ and $5$ there exist infinitely many multiples of $p$ of the form $1111...1$.

2011 Brazil Team Selection Test, 2

Tags: prime , number theory

Let $n\ge 3$ be an integer such that for every prime factor $q$ of $n-1$ exists an integer $a > 1$ such that $a^{n-1} \equiv 1 \,(\mod n \, )$ and $a^{\frac{n-1} {q}}\not\equiv 1 \,(\mod n \, )$. Prove that $n$ is not prime.

2016 JBMO Shortlist, 1

Tags: prime , number theory

Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.

1997 Israel National Olympiad, 5

Tags: prime , number theory , coprime

The natural numbers $a_1,a_2,...,a_n, n \ge 12$, are smaller than $9n^2$ and pairwise coprime. Show that at least one of these numbers is prime.

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.

2015 Saudi Arabia BMO TST, 4

Tags: prime , coprime , gcd , number theory

Let $n \ge 2$ be an integer and $p_1 < p_2 < ... < p_n$ prime numbers. Prove that there exists an integer $k$ relatively prime with $p_1p_2... p_n$ and such that $gcd (k + p_1p_2...p_i, p_1p_2...p_n) = 1$ for all $i = 1, 2,..., n - 1$. Malik Talbi

2012 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: prime , number theory

Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$

2013 Thailand Mathematical Olympiad, 1

Tags: prime , divide , number theory

Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$

2019 Argentina National Olympiad, 5

Tags: prime , arithmetic sequence , number theory

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 $d$.

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , remainder , prime

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

2010 Thailand Mathematical Olympiad, 10

Tags: binomial , prime , number theory , divisible

Find all primes $p$ such that ${100 \choose p} + 7$ is divisible by $p$.

2002 Croatia Team Selection Test, 3

Tags: prime , number theory

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

2014 IFYM, Sozopol, 2

Tags: prime , point , number theory , circles

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 $r$.

1997 Israel Grosman Mathematical Olympiad, 1

Tags: prime , digit , number theory

Prove that there are at most three primes between $10$ and $10^{10}$ all of whose decimal digits are $1$.

Oliforum Contest V 2017, 4

Tags: prime , number theory

Let $p_n$ be the $n$-th prime, so that $p_1 = 2, p_2 = 3,...$ and define $$X_n = \{0\} \cup \{p_1,...,p_n\}$$ for each positive integer $n$. Find all $n$ for which there exist $A,B \subseteq N$ such that$ |A|,|B| \ge 2$ and $$X_n = A + B$$, where $A + B :=\{a + b : a \in A; b \in B \}$ and $N := \{0,1, 2,...\}$. (Salvatore Tringali)

1999 Romania National Olympiad, 2

Tags: prime , number theory

Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$ Prove that $a^2 +b^2 +c^2$ cannot be a prime number.

2010 QEDMO 7th, 1

Tags: prime , number theory

Find all natural numbers $n$ for which both $n^n + 1$ and $(2n)^{2n} + 1$ are prime numbers.

2023 Brazil Team Selection Test, 3

Tags: prime factorization , prime number , prime , function , number theory

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2003 IMO Shortlist, 7

Tags: recurrence , sequence , divisibility , modular arithmetic , prime , polynomial

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

1997 Estonia National Olympiad, 1

Tags: prime , composite , number theory

Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.