Olympiad problems

2003 France Team Selection Test, 3

Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2005 Korea Junior Math Olympiad, 6

Tags: set theory , prime , number theory

For two different prime numbers $p, q$, define $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.

2020 Argentina National Olympiad, 4

Tags: prime , integer , number theory

Let $a$ and $b$ be positive integers such that $\frac{5a^4 + a^2}{b^4 + 3b^2 + 4}$ is an integer. Show that $a$ is not prime.

2012 IMO Shortlist, N8

Tags: gauss , modular arithmetic , number theory , divisibility , prime

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

2013 Greece JBMO TST, 3

Tags: diophantine equation , number theory , prime

If $p$ is a prime positive integer and $x,y$ are positive integers, find , in terms of $p$, all pairs $(x,y)$ that are solutions of the equation: $p(x-2)=x(y-1)$. (1) If it is also given that $x+y=21$, find all triplets $(x,y,p)$ that are solutions to equation (1).

2017 Israel Oral Olympiad, 3

Tags: number theory , prime , prime number

2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2019 AMC 10, 2

Tags: prime

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

2019 Saudi Arabia Pre-TST + Training Tests, 5.1

Tags: number theory , prime , divide , divisible

Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

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 $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

2017 Czech-Polish-Slovak Junior Match, 1

Tags: perfect cube , number theory , prime

Decide if there are primes $p, q, r$ such that $(p^2 + p) (q^2 + q) (r^2 + r)$ is a square of an integer.

1977 Kurschak Competition, 1

Tags: composite , prime , number theory

Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$.

2017 International Zhautykov Olympiad, 2

Tags: number theory , divisor , prime

For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.

2008 IMAC Arhimede, 1

Tags: number theory , prime , perfect square

Find all prime numbers $ p $ for which $ 1 + p\cdot 2^{p} $ is a perfect square.

2024 Kosovo Team Selection Test, P1

Tags: number theory , prime

Find all prime numbers $p$ and $q$ such that $p^q + 5q - 2$ is also a prime number.

1992 All Soviet Union Mathematical Olympiad, 567

Tags: number theory , coprime , prime

Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.

2018 Saudi Arabia IMO TST, 1

Tags: prime divisor , theory , divisor , prime

Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

1995 Korea National Olympiad, Day 2

Tags: polynomial , integer polynomial , greatest common divisor , prime , factorization , algebra

Let $a,b$ be integers and $p$ be a prime number such that: (i) $p$ is the greatest common divisor of $a$ and $b$; (ii) $p^2$ divides $a$. Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.

2020 Regional Olympiad of Mexico Northeast, 4

Tags: divide , prime , perfect square , number theory

Let $n > 1$ be an integer and $p$ be a prime. Prove that if $n|p-1$ and $p|n^3-1$, then $4p-3$ is a perfect square.

2016 Mathematical Talent Reward Programme, MCQ: P 4

Tags: prime

There are 168 primes below 1000. Then sum of all primes below 1000 is [list=1] [*] 11555 [*] 76127 [*] 57298 [*] 81722 [/list]

2010 Thailand Mathematical Olympiad, 10

Tags: number theory , binomial , divisible , prime

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

2017 Junior Regional Olympiad - FBH, 4

Tags: statements , divisible , prime

Let $n$ and $k$ be positive integers for which we have $4$ statements: $i)$ $n+1$ is divisible with $k$ $ii)$ $n=2k+5$ $iii)$ $n+k$ is divisible with $3$ $iv)$ $n+7k$ is prime Determine all possible values for $n$ and $k$, if out of the $4$ statements, three of them are true and one is false

2022 239 Open Mathematical Olympiad, 4

Tags: number theory , natural number , prime

Vasya has a calculator that works with pairs of numbers. The calculator knows hoe to make a pair $(x+y,x)$ or a pair $(2x+y+1,x+y+1)$ from a pair $(x,y).$ At the beginning, the pair $(1,1)$ is presented on the calculator. Prove that for any natural $n$ there is exactly one pair $(n,k)$ that can be obtained using a calculator.

2006 MOP Homework, 5

Tags: prime , sum , number theory

Let $n$ be a nonnegative integer, and let $p$ be a prime number that is congruent to $7$ modulo $8$. Prove that $$\sum_{k=1}^{p} \left\{ \frac{k^{2n}}{p} - \frac{1}{2} \right\} = \frac{p-1}{2}$$

2005 Austrian-Polish Competition, 4

Tags: perfect square , fermat's little theorem , prime , number theory , diophantine equation

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

2021 Nordic, 1

Tags: number theory , prime

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.