Olympiad problems

2021 Bangladeshi National Mathematical Olympiad, 1

How many ordered pairs of integers $(m,n)$ are there such that $m$ and $n$ are the legs of a right triangle with an area equal to a prime number not exceeding $80$?

2016 Bundeswettbewerb Mathematik, 2

Tags: number theory , prime numbers , infinitely many solutions

Prove that there are infinitely many positive integers that cannot be expressed as the sum of a triangular number and a prime number.

2019 JBMO Shortlist, N1

Tags: number theory , Junior , JBMO , Balkan , 2019 , prime numbers

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

1973 Putnam, B3

Tags: Putnam , number theory , prime numbers , polynomial

Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions: $$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$

Russian TST 2015, P1

Tags: number theory , prime numbers

Prove that there exist two natural numbers $a,b$ such that $|a-m|+|b-n|>1000$ for any relatively prime natural numbers $m,n$.

1978 IMO Longlists, 10

Tags: number theory , prime numbers , arithmetic sequence , number theory proposed

Show that for any natural number $n$ there exist two prime numbers $p$ and $q, p \neq q$, such that $n$ divides their difference.

2010 Contests, 1

Tags: number theory , prime numbers , number theory proposed

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?

1992 Spain Mathematical Olympiad, 4

Tags: number theory , prime numbers , arithmetic sequence

Prove that the arithmetic progression $3,7,11,15,...$. contains infinitely many prime numbers.

2024 Thailand October Camp, 4

Tags: number theory , prime numbers

The sequence $(a_n)_{n\in\mathbb{N}}$ is defined by $a_1=3$ and $$a_n=a_1a_2\cdots a_{n-1}-1$$ Show that there exist infinitely many prime number that divide at least one number in this sequences

2001 Bundeswettbewerb Mathematik, 4

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

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

2018 Centroamerican and Caribbean Math Olympiad, 3

Tags: number theory , Centroamerican , prime numbers

Let $x, y$ be real numbers such that $x-y, x^2-y^2, x^3-y^3$ are all prime numbers. Prove that $x-y=3$. EDIT: Problem submitted by Leonel Castillo, Panama.

2019 Polish MO Finals, 2

Tags: number theory , prime numbers , Divisibility

Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.

1991 AIME Problems, 5

Tags: number theory , prime numbers , prime factorization

Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1 will $ 20!$ be the resulting product?

2017 Iran MO (3rd round), 1

Tags: number theory , Divisibility , prime numbers , prime , Iran , IranMO

Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then $$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$ is not an integer.

2019 Switzerland Team Selection Test, 2

Tags: number theory , prime numbers

Find the largest prime $p$ such that there exist positive integers $a,b$ satisfying $$p=\frac{b}{2}\sqrt{\frac{a-b}{a+b}}.$$

2023 VN Math Olympiad For High School Students, Problem 6

Tags: algebra , polynomial , number theory , prime numbers

Prove that these polynomials are irreducible in $\mathbb{Q}[x]:$ a) $\frac{{{x^p}}}{{p!}} + \frac{{{x^{p - 1}}}}{{(p - 1)!}} + ... + \frac{{{x^2}}}{2} + x + 1,$ with $p$ is a prime number. b) $x^{2^n}+1,$ with $n$ is a positive integer.

2005 Iran MO (3rd Round), 5

Tags: induction , number theory , prime numbers , number theory proposed

Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.

2021 Moldova Team Selection Test, 2

Tags: number theory , prime numbers

Prove that if $p$ and $q$ are two prime numbers, such that $$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$ then $p=q$.

2013 Bangladesh Mathematical Olympiad, 7

Tags: number theory , prime numbers

Higher Secondary P7 If there exists a prime number $p$ such that $p+2q$ is prime for all positive integer $q$ smaller than $p$, then $p$ is called an "awesome prime". Find the largest "awesome prime" and prove that it is indeed the largest such prime.

2020 AMC 10, 4

Tags: AMC , AMC 10 , prime numbers , 2020 AMC , 2020 AMC 10B , 2020 AMC 12B , AMC 12

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

2007 ITest, 1

Tags: number theory , prime numbers

A twin prime pair is a pair of primes $(p,q)$ such that $q = p + 2$. The Twin Prime Conjecture states that there are infinitely many twin prime pairs. What is the arithmetic mean of the two primes in the smallest twin prime pair? (1 is not a prime.) $\textbf{(A) }4$

2007 Bulgarian Autumn Math Competition, Problem 12.4

Tags: prime numbers , number theory , linear recurrence

Let $p$ and $q$ be prime numbers and $\{a_{n}\}_{n=1}^{\infty}$ be a sequence of integers defined by: \[a_{0}=0, a_{1}=1, a_{n+2}=pa_{n+1}-qa_{n}\quad\forall n\geq 0\] Find $p$ and $q$ if there exists an integer $k$ such that $a_{3k}=-3$.

2018 Israel Olympic Revenge, 1

Tags: prime numbers , Divisibility , number theory , Divisors

Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

2004 Iran MO (3rd Round), 1

Tags: number theory , prime numbers , combinatorics proposed , combinatorics

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

2010 Pan African, 1

Tags: number theory , prime numbers , number theory proposed

a) Show that it is possible to pair off the numbers $1,2,3,\ldots ,10$ so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers $1,2,3,\ldots ,20$ so that the sums of each of the ten pairs are ten different prime numbers?