Olympiad problems

1973 AMC 12/AHSME, 18

If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder $ \textbf{(A)}\ \text{never} \qquad \textbf{(B)}\ \text{sometimes only} \qquad \textbf{(C)}\ \text{always} \qquad$ $ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad \textbf{(E)}\ \text{none of these}$

2016 India IMO Training Camp, 3

Tags: number theory , set , prime number

Let $\mathbb N$ denote the set of all natural numbers. Show that there exists two nonempty subsets $A$ and $B$ of $\mathbb N$ such that [list=1] [*] $A\cap B=\{1\};$ [*] every number in $\mathbb N$ can be expressed as the product of a number in $A$ and a number in $B$; [*] each prime number is a divisor of some number in $A$ and also some number in $B$; [*] one of the sets $A$ and $B$ has the following property: if the numbers in this set are written as $x_1<x_2<x_3<\cdots$, then for any given positive integer $M$ there exists $k\in \mathbb N$ such that $x_{k+1}-x_k\ge M$. [*] Each set has infinitely many composite numbers. [/list]

2015 Benelux, 3

Tags: number theory , prime number

Does there exist a prime number whose decimal representation is of the form $3811\cdots11$ (that is, consisting of the digits $3$ and $8$ in that order, followed by one or more digits $1$)?

2019 Hong Kong TST, 1

Tags: number theory , prime number

Determine all sequences $p_1, p_2, \dots $ of prime numbers for which there exists an integer $k$ such that the recurrence relation \[ p_{n+2} = p_{n+1} + p_n + k \] holds for all positive integers $n$.

2016 Hong Kong TST, 4

Tags: number theory , prime number

Mable and Nora play a game according to the following steps in order: 1. Mable writes down any 2015 distinct prime numbers in ascending order in a row. The product of these primes is Marble's score. 2. Nora writes down a positive integer 3. Mable draws a vertical line between two adjacent primes she has written in step 1, and compute the product of the prime(s) on the left of the vertical line 4. Nora must add the product obtained by Marble in step 3 to the number she has written in step 2, and the sum becomes Nora's score. If Marble and Nora's scores have a common factor greater than 1, Marble wins, otherwise Nora wins. Who has a winning strategy?

2012-2013 SDML (Middle School), 10

Tags: number theory , prime number

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

2000 AMC 12/AHSME, 6

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$

2017 Poland - Second Round, 6

Tags: prime number , number theory

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 $x = y$.

2014 AMC 8, 23

Tags: number theory , prime number

Three members of the Euclid Middle School girls' softball team had the following conversation. Ashley: I just realized that our uniform numbers are all $2$-digit primes. Bethany: And the sum of your two uniform numbers is the date of my birthday earlier this month. Caitlin: That's funny. The sum of your two uniform numbers is the date of my birthday later this month. Ashley: And the sum of you two uniform numbers is today's date. What number does Caitlin wear? $\textbf{(A) }11\qquad\textbf{(B) }13\qquad\textbf{(C) }17\qquad\textbf{(D) }19\qquad \textbf{(E) }23$

2008 Tuymaada Olympiad, 2

Tags: prime number , number theory

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

2009 Princeton University Math Competition, 3

Tags: prime number , number theory

Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

2021 2nd Memorial "Aleksandar Blazhevski-Cane", 2

Tags: number theory , prime number

Let $p$ be a prime number and $F=\left \{0,1,2,...,p-1 \right \}$. Let $A$ be a proper subset of $F$ that satisfies the following property: if $a,b \in A$, then $ab+1$ (mod $p$) $ \in A$. How many elements can $A$ have? (Justify your answer.)

2006 Germany Team Selection Test, 1

Tags: algebra , polynomial , number theory , 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.

2023 Junior Macedonian Mathematical Olympiad, 2

Tags: number theory , prime number

A positive integer is called [i]superprime[/i] if the difference between any two of its consecutive positive divisors is a prime number. Determine all superprime integers. [i]Authored by Nikola Velov[/i]

2004 Iran MO (3rd Round), 1

Tags: number theory , prime number , 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.

1997 Singapore Team Selection Test, 1

Tags: number theory , prime number , game , invariant

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

2006 Turkey Team Selection Test, 1

Tags: induction , quadratic , gauss , modular arithmetic , 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 JBMO Shortlist, 4

Tags: shortlist , number theory , prime number

Find all prime numbers $p$ such that $(x + y)^{19} - x^{19} - y^{19}$ is a multiple of $p$ for any positive integers $x$, $y$.

2019 China Western Mathematical Olympiad, 7

Tags: number theory , prime number

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

2016 Spain Mathematical Olympiad, 2

Tags: number theory , prime number , quadratic residue

Given a positive prime number $p$. Prove that there exist a positive integer $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist a positive integer $\beta$ such that $p|\beta(\beta-1)+25$.

2016 Uzbekistan National Olympiad, 2

Tags: number theory , prime number

$n$ is natural number and $p$ is prime number. If $1+np$ is square of natural number then prove that $n+1$ is equal to some sum of $p$ square of natural numbers.

2018 Brazil Team Selection Test, 2

Tags: number theory , relatively prime , prime number

Prove that there is an integer $n>10^{2018}$ such that the sum of all primes less than $n$ is relatively prime to $n$. [i](R. Salimov)[/i]

2019 Romania National Olympiad, 4

Tags: prime number , group theory , linear algebra , matrix , permutation

Let $p$ be a prime number. For any $\sigma \in S_p$ (the permutation group of $\{1,2,...,p \}),$ define the matrix $A_{\sigma}=(a_{ij}) \in \mathcal{M}_p(\mathbb{Z})$ as $a_{ij} = \sigma^{i-1}(j),$ where $\sigma^0$ is the identity permutation and $\sigma^k = \underbrace{\sigma \circ \sigma \circ ... \circ \sigma}_k.$ Prove that $D = \{ |\det A_{\sigma}| : \sigma \in S_p \}$ has at most $1+ (p-2)!$ elements.

2023 Grosman Mathematical Olympiad, 4

Tags: quadratic , number theory , prime number

Let $q$ be an odd prime number. Prove that it is impossible for all $(q-1)$ numbers \[1^2+1+q, 2^2+2+q, \dots, (q-1)^2+(q-1)+q\] to be products of two primes (not necessarily distinct).

2019 SG Originals, Q4

Tags: number theory , prime number , algebra , polynomial

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]