Olympiad problems

2007 Thailand Mathematical Olympiad, 18

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

1990 IMO Longlists, 98

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.

2011 ISI B.Stat Entrance Exam, 7

Tags: arithmetic sequence , number theory , prime number

[b](i)[/b] Show that there cannot exists three peime numbers, each greater than $3$, which are in arithmetic progression with a common difference less than $5$. [b](ii)[/b] Let $k > 3$ be an integer. Show that it is not possible for $k$ prime numbers, each greater than $k$, to be in an arithmetic progression with a common difference less than or equal to $k+1$.

2012 IMO Shortlist, N3

Tags: binomial coefficient , number theory , prime number

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

VMEO IV 2015, 12.3

Tags: number theory , arithmetic sequence , prime number

Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

2022 USAMO, 4

Tags: number theory , diophantine equation , prime number

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

2018 IFYM, Sozopol, 3

Tags: number theory , prime number

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

2012 JBMO ShortLists, 2

Tags: number theory , prime number

Do there exist prime numbers $p$ and $q$ such that $p^2(p^3-1)=q(q+1)$ ?

2004 China Western Mathematical Olympiad, 4

Tags: logarithm , number theory , search , function , euler , totient function , prime number

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$. Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$, smaller than $n$). Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that \[ d(n) + \varphi(n) = n+c , \] and for such $c$ find all values of $n$ satisfying the above relationship.

2016 Bundeswettbewerb Mathematik, 1

Tags: number theory , prime number , decimal representation

A number with $2016$ zeros that is written as $101010 \dots 0101$ is given, in which the zeros and ones alternate. Prove that this number is not prime.

2023 India IMO Training Camp, 3

Tags: function , number theory , prime , prime factorization , prime number

Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$. (For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)

2010 India IMO Training Camp, 8

Tags: prime number , number theory

Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

2024 Kyiv City MO Round 1, Problem 2

Tags: number theory , prime number

Write the numbers from $1$ to $16$ in the cells of a of a $4 \times 4$ square so that: 1. Each cell contains exactly one number; 2. Each number is written exactly once; 3. For any two cells that are symmetrical with respect to any of the perpendicular bisectors of sides of the original $4 \times 4$ square, the sum of numbers in them is a prime number The figure below shows examples of such pairs of cells, sums of numbers in which have to be prime. [img]https://i.ibb.co/fqX05dY/Kyiv-MO-2024-Round-1-8-2.png[/img] [i]Proposed by Mykhailo Shtandenko[/i]

2015 IFYM, Sozopol, 1

Tags: prime number , divisibility , number theory

Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

2014 Romania National Olympiad, 4

Tags: number theory , prime number , group theory , abstract algebra

Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $ [b]a)[/b] Prove that the order of $ G $ is a power of $ p. $ [b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $

2020 Hong Kong TST, 4

Tags: floor function , prime number , number theory

Find the total number of primes $p<100$ such that $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$ is divisible by $p$. Here $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$.

2018 Polish Junior MO First Round, 3

Tags: number theory , prime number , divisibility

Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.

2015 Turkey Junior National Olympiad, 3

Tags: number theory , diophantine equation , prime number

Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds.

2010 Brazil National Olympiad, 2

Tags: number theory , algebra , polynomial , prime number , ceiling function

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

2014 JBMO Shortlist, 4

Tags: number theory , prime number

Prove that there are not intgers $a$ and $b$ with conditions, i) $16a-9b$ is a prime number. ii) $ab$ is a perfect square. iii) $a+b$ is also perfect square.

2014 Contests, 1

Tags: function , number theory , prime number

A positive proper divisor is a positive divisor of a number, excluding itself. For positive integers $n \ge 2$, let $f(n)$ denote the number that is one more than the largest proper divisor of $n$. Determine all positive integers $n$ such that $f(f(n)) = 2$.

2012 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , prime number , prime

Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime

1998 AMC 12/AHSME, 12

Tags: number theory , prime number , logarithm

How many different prime numbers are factors of $ N$ if \[ \log_2 (\log_3 (\log_5 (\log_7 N))) \equal{} 11? \]$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 7$

2021 Kyiv City MO Round 1, 11.5

Tags: number theory , prime number

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

1998 India Regional Mathematical Olympiad, 2

Tags: modular arithmetic , prime number , number theory

Let $n$ be a positive integer and $p_1, p_2, p_3, \ldots p_n$ be $n$ prime numbers all larger than $5$ such that $6$ divides $p_1 ^2 + p_2 ^2 + p_3 ^2 + \cdots p_n ^2$. prove that $6$ divides $n$.