Olympiad problems

2017 Poland - Second Round, 6

A prime number $p > 2$ and $x,y \in \left\{ 1,2,\ldots, \frac{p-1}{2} \right\}$ are given. Prove that if $x\left( p-x\right)y\left( p-y\right)$ is a perfect square, then $x = y$.

2023 Singapore Junior Math Olympiad, 4

Tags: number theory , prime number

Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.

1990 IMO Shortlist, 27

Tags: number theory , decimal representation , prime number , digit

Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.

2019 Junior Balkan MO, 1

Tags: number theory , prime number

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.

2012 European Mathematical Cup, 2

Tags: function , number theory , greatest common divisor , prime number

Let $S$ be the set of positive integers. For any $a$ and $b$ in the set we have $GCD(a, b)>1$. For any $a$, $b$ and $c$ in the set we have $GCD(a, b, c)=1$. Is it possible that $S$ has $2012$ elements? [i]Proposed by Ognjen Stipetić.[/i]

2017 Brazil Undergrad MO, 2

Tags: number theory , prime number , sequence

Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.

1999 AMC 12/AHSME, 4

Tags: number theory , prime number

Find the sum of all prime numbers between $ 1$ and $ 100$ that are simultaneously $ 1$ greater than a multiple of $ 4$ and $ 1$ less than a multiple of $ 5$. $ \textbf{(A)}\ 118\qquad \textbf{(B)}\ 137\qquad \textbf{(C)}\ 158\qquad \textbf{(D)}\ 187 \qquad \textbf{(E)}\ 245$

2010 Slovenia National Olympiad, 1

Tags: prime number , number theory

Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.

1994 China Team Selection Test, 2

Tags: algebra , polynomial , calculus , integration , gauss , prime number , number theory

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

1982 Bulgaria National Olympiad, Problem 1

Tags: number theory , inequalities , prime number

Find all pairs of natural numbers $(n,k)$ for which $(n+1)^{k}-1 = n!$.

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , prime number

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

2024 Chile Classification NMO Seniors, 3

Tags: combinatorics , prime number

Is it possible to place 100 consecutive numbers around a circle in some order such that the product of each pair of adjacent numbers is always a perfect square? (Recall that a number is a perfect square if it is the square of an integer.)

2000 AMC 10, 11

Tags: number theory , prime number

Two different prime numbers between $ 4$ and $ 18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $ \textbf{(A)}\ 21 \qquad \textbf{(B)}\ 60\qquad \textbf{(C)}\ 119 \qquad \textbf{(D)}\ 180\qquad \textbf{(E)}\ 231$

PEN E Problems, 35

Tags: number theory , prime number

There exists a block of $1000$ consecutive positive integers containing no prime numbers, namely, $1001!+2$, $1001!+3$, $\cdots$, $1001!+1001$. Does there exist a block of $1000$ consecutive positive integers containing exactly five prime numbers?

2006 Germany Team Selection Test, 1

Tags: polynomial , number theory , algebra , composite number , prime number

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.

2019 Moldova Team Selection Test, 12

Tags: number theory , prime number

Let $p\ge 5$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n\le \frac{p+1}{2}$ for which $p$ divides $2^n\cdot 3^m-1.$

2011 Philippine MO, 3

Tags: combinatorial geometry , number theory , prime number

The $2011$th prime number is $17483$ and the next prime is $17489$. Does there exist a sequence of $2011^{2011}$ consecutive positive integers that contain exactly $2011$ prime numbers?

2016 Croatia Team Selection Test, Problem 4

Tags: number theory , prime , diophantine equation , prime number

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

2006 Turkey Team Selection Test, 1

Tags: induction , modular arithmetic , quadratic , gauss , prime number , number theory

For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

2020 Kosovo National Mathematical Olympiad, 3

Tags: number theory , prime number , olym

Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.

2004 Hong kong National Olympiad, 4

Tags: function , combinatorics , number theory , prime number

Let $S=\{1,2,...,100\}$ . Find number of functions $f: S\to S$ satisfying the following conditions a)$f(1)=1$ b)$f$ is bijective c)$f(n)=f(g(n))f(h(n))\forall n\in S$, where $g(n),h(n)$ are positive integer numbers such that $g(n)\leq h(n),n=g(n)h(n)$ that minimize $h(n)-g(n)$.

2016 China Team Selection Test, 4

Tags: sequence , prime number , number theory

Let $c,d \geq 2$ be naturals. Let $\{a_n\}$ be the sequence satisfying $a_1 = c, a_{n+1} = a_n^d + c$ for $n = 1,2,\cdots$. Prove that for any $n \geq 2$, there exists a prime number $p$ such that $p|a_n$ and $p \not | a_i$ for $i = 1,2,\cdots n-1$.

2022 Korea Junior Math Olympiad, 3

Tags: number theory , prime number

For a given odd prime number $p$, define $f(n)$ the remainder of $d$ divided by $p$, where $d$ is the biggest divisor of $n$ which is not a multiple of $p$. For example when $p=5$, $f(6)=1, f(35)=2, f(75)=3$. Define the sequence $a_1, a_2, \ldots, a_n, \ldots$ of integers as the followings: [list] [*]$a_1=1$ [*]$a_{n+1}=a_n+(-1)^{f(n)+1}$ for all positive integers $n$. [/list] Determine all integers $m$, such that there exist infinitely many positive integers $k$ such that $m=a_k$.

2006 ISI B.Stat Entrance Exam, 3

Tags: modular arithmetic , number theory , induction , prime number

Prove that $n^4 + 4^{n}$ is composite for all values of $n$ greater than $1$.

2019 Singapore MO Open, 4

Tags: number theory , prime number , polynomial , algebra

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]