Olympiad problems

2008 IMAR Test, 2

A point $ P$ of integer coordinates in the Cartesian plane is said [i]visible[/i] if the segment $ OP$ does not contain any other points with integer coordinates (except its ends). Prove that for any $ n\in\mathbb{N}^*$ there exists a visible point $ P_{n}$, at distance larger than $ n$ from any other visible point. [b]Dan Schwarz[/b]

2011 Greece Team Selection Test, 1

Tags: number theory , prime numbers

Find all prime numbers $p,q$ such that: $$p^4+p^3+p^2+p=q^2+q$$

1973 AMC 12/AHSME, 18

Tags: probability , modular arithmetic , number theory , prime numbers

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}$

2015 Miklos Schweitzer, 4

Tags: number theory , number theory proposed , limits , limit , algebra and number theory , prime numbers

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$, the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$. Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$.

2022 Romania EGMO TST, P4

Tags: number theory , romania , EGMO , prime numbers

Let $p\geq 3$ be an odd positive integer. Show that $p$ is prime if and only if however we choose $(p+1)/2$ pairwise distinct positive integers, we can find two of them, $a$ and $b$, such that $(a+b)/\gcd(a,b)\geq p.$

2023 Bulgaria JBMO TST, 3

Tags: number theory , prime numbers , Diophantine equation

Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that: $\blacksquare$ $4\nmid c$ $\blacksquare$ $p\not\equiv 11\pmod{16}$ $\blacksquare$ $p^aq^b-1=(p+4)^c$

2023 Silk Road, 3

Tags: number theory , prime numbers , Quadratic Residues , Directed graphs , quadratics

Let $p$ be a prime number. We construct a directed graph of $p$ vertices, labeled with integers from $0$ to $p-1$. There is an edge from vertex $x$ to vertex $y$ if and only if $x^2+1\equiv y \pmod{p}$. Let $f(p)$ denotes the length of the longest directed cycle in this graph. Prove that $f(p)$ can attain arbitrarily large values.

2021 Centroamerican and Caribbean Math Olympiad, 1

Tags: number theory , prime numbers

An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.

2018 AMC 12/AHSME, 5

Tags: AMC , AMC 10 , AMC 10 B , number theory , prime numbers , 2018 AMC 10B

How many subsets of $\{2,3,4,5,6,7,8,9\}$ contain at least one prime number? $\textbf{(A)} \text{ 128} \qquad \textbf{(B)} \text{ 192} \qquad \textbf{(C)} \text{ 224} \qquad \textbf{(D)} \text{ 240} \qquad \textbf{(E)} \text{ 256}$

1990 Irish Math Olympiad, 2

Tags: number theory , prime numbers , arithmetic sequence

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.

2022 USAMO, 4

Tags: number theory , Diophantine equation , prime numbers

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

2019 Dutch IMO TST, 4

Tags: number theory , function , Find all functions , prime numbers , functional equation

Find all functions $f : Z \to Z$ satisfying $\bullet$ $ f(p) > 0$ for all prime numbers $p$, $\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.

2014 Poland - Second Round, 6.

Tags: number theory , prime numbers

Call a positive number $n$ [i]fine[/i], if there exists a prime number $p$ such that $p|n$ and $p^2\nmid n$. Prove that at least 99% of numbers $1, 2, 3, \ldots, 10^{12}$ are fine numbers.

2013 Ukraine Team Selection Test, 3

Tags: algebra , number theory , IMO Shortlist , prime numbers , polynomial

For a nonnegative integer $n$ define $\operatorname{rad}(n)=1$ if $n=0$ or $n=1$, and $\operatorname{rad}(n)=p_1p_2\cdots p_k$ where $p_1<p_2<\cdots <p_k$ are all prime factors of $n$. Find all polynomials $f(x)$ with nonnegative integer coefficients such that $\operatorname{rad}(f(n))$ divides $\operatorname{rad}(f(n^{\operatorname{rad}(n)}))$ for every nonnegative integer $n$.

PEN E Problems, 35

Tags: number theory , prime numbers

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?

2005 IMO Shortlist, 5

Tags: number theory , prime numbers , prime factorization , IMO Shortlist

Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$. Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$. (a) Show that there are only finitely many pairs of consecutive highly divisible integers of the form $(a, b)$ with $a\mid b$. (b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.

2004 France Team Selection Test, 3

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

Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$. Prove that $M = P$.

2003 SNSB Admission, 3

Tags: number theory , prime numbers , Ring Theory , factorial

Let be a prime number $ p, $ the quotient ring $ R=\mathbb{Z}[X,Y]/(pX,pY), $ and a prime ideal $ I\supset pA $ that is not maximal. Show that the ring $ \left\{ r/i|r\in R, i\in I \right\} $ is factorial.

2022 Singapore MO Open, Q5

Tags: number theory , polynomial , modulo , algebra , prime numbers

Let $n\ge 2$ be a positive integer. For any integer $a$, let $P_a(x)$ denote the polynomial $x^n+ax$. Let $p$ be a prime number and define the set $S_a$ as the set of residues mod $p$ that $P_a(x)$ attains. That is, $$S_a=\{b\mid 0\le b\le p-1,\text{ and there is }c\text{ such that }P_a(c)\equiv b \pmod{p}\}.$$Show that the expression $\frac{1}{p-1}\sum\limits_{a=1}^{p-1}|S_a|$ is an integer. [i]Proposed by fattypiggy123[/i]

2024 CAPS Match, 6

Tags: combinatorics , number theory , international competitions , Combinatorial Number Theory , prime numbers , triplets , positive integers

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

2013 Dutch IMO TST, 3

Tags: number theory , prime numbers , number theory unsolved

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

2014 CentroAmerican, 1

Tags: number theory , prime numbers , number theory proposed

A positive integer is called [i]tico[/i] if it is the product of three different prime numbers that add up to 74. Verify that 2014 is tico. Which year will be the next tico year? Which one will be the last tico year in history?

2005 iTest, 26

Tags: number theory , prime numbers

Joe and Kathryn are both on the school math team, which practices every Wednesday after school until $4$ PM for competitions. The team was preparing for the $ 2005$ iTest when Joe realized how crazy he was for not asking Kathryn out – the way she worked those iTest problems, solving question after question, almost made him go insane sitting there that day. He never felt the same way when she worked on preparing for other competitions – they just aren’t the same. Kathryn always beat Joe at competitions, too. Joe admired her resolve and unwillingness to make herself look stupid, when so many other girls he knew at school tried to pretend they were stupid in order to attract guys. So as time ticked away and that afternoon’s Wednesday practice neared an end, Joe was determined to strike up a conversation with Kathryn and ask her out. He really wanted to impress her, so he thought he’d ask her a really hard history of math question that she didn’t know. Naturally, she’d want the answer, and be so impressed with Joe’s brilliance that she’d go out with him on Friday night. Great plan. Seriously. When Joe asked Kathryn after class, “Who was the mathematician that died in approximately $200$ B.C. that developed a method for calculating all prime numbers?” Kathryn gave the correct response. What name did she say?

2022 Assara - South Russian Girl's MO, 5

Tags: number theory , prime numbers , prime

Find all pairs of prime numbers $p, q$ such that the number $pq + p - 6$ is also prime.

Kvant 2020, M2597

Tags: number theory , Kvant , prime numbers

Let $p{}$ be a prime number greater than 3. Prove that there exists a natural number $y{}$ less than $p/2$ and such that the number $py + 1$ cannot be represented as a product of two integers, each of which is greater than $y{}$. [i]Proposed by M. Antipov[/i]