Olympiad problems

PEN J Problems, 3

If $p$ is a prime and $n$ an integer such that $1<n \le p$, then \[\phi \left( \sum_{k=0}^{p-1}n^{k}\right) \equiv 0 \; \pmod{p}.\]

2022 Purple Comet Problems, 23

Tags: algebra , number theory

There are prime numbers $a$, $b$, and $c$ such that the system of equations $$a \cdot x - 3 \cdot y + 6 \cdot z = 8$$ $$b \cdot x + 3\frac12 \cdot y + 2\frac13 \cdot z = -28$$ $$c \cdot x - 5\frac12 \cdot y + 18\frac13 \cdot z = 0$$ has infinitely many solutions for $(x, y, z)$. Find the product $a \cdot b \cdot c$.

2011 Kosovo Team Selection Test, 4

Tags: number theory , modular arithmetic

From the number $7^{1996}$ we delete its first digit, and then add the same digit to the remaining number. This process continues until the left number has ten digits. Show that the left number has two same digits.

2022 Romania EGMO TST, P4

Tags: number theory , function

For every positive integer $N\geq 2$ with prime factorisation $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ we define \[f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.\] Let $x_0\geq 2$ be a positive integer. We define the sequence $x_{n+1}=f(x_n)$ for all $n\geq 0.$ Prove that this sequence is eventually periodic and determine its fundamental period.

2016 Belarus Team Selection Test, 3

Tags: number theory , factorial , divisibility

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

2000 France Team Selection Test, 3

Tags: greatest common divisor , number theory

Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.

2012 AMC 12/AHSME, 14

Tags: number theory , inequalities , modular arithmetic

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divide

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2020 HMNT (HMMO), 1

Tags: number theory

Chelsea goes to La Verde's at MIT and buys $100$ coconuts, each weighing $4$ pounds, and $100$ honeydews, each weighing $5$ pounds. She wants to distribute them among $n$ bags, so that each bag contains at most $13$ pounds of fruit. What is the minimum $n$ for which this is possible?

2016 Postal Coaching, 5

Tags: number theory , set theory , group theory , binary operation

Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds; [*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]

2005 France Team Selection Test, 4

Tags: inequalities , number theory , induction , floor function , logarithm , ceiling function

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

2010 Indonesia TST, 2

Tags: number theory , greatest common divisor , sequence

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]

2010 Indonesia TST, 4

Tags: number theory

Let $n$ be a positive integer with $n = p^{2010}q^{2010}$ for two odd primes $p$ and $q$. Show that there exist exactly $\sqrt[2010]{n}$ positive integers $x \le n$ such that $p^{2010}|x^p - 1$ and $q^{2010}|x^q - 1$.

2018 CMIMC Number Theory, 3

Tags: number theory

Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are both in $S$.

2012 Online Math Open Problems, 11

Tags: number theory , relatively prime

If \[\frac{1} {x} + \frac{1} {2x^2} +\frac{1} {4x^3}+\frac{1}{8x^4}+\frac{1}{16x^5}+\cdots=\frac{1} {64}, \] and $x$ can be expressed in the form $\frac{m}{n},$ where $m,n$ are relatively prime positive integers, find $m+n$. [i]Author: Ray Li[/i]

KoMaL A Problems 2023/2024, A. 882

Tags: number theory , greatest common divisor , gcd

Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that \[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\] [i]Proposed by Dávid Matolcsi, Berkeley[/i]

2010 Contests, 2

Tags: induction , inequalities , prime factorization , number theory

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

2022 South East Mathematical Olympiad, 7

Tags: number theory

Prove that for any positive real number $\lambda$,there are $n$ positive numbers $a_1,a_2,\cdots,a_n(n\geq 2)$,so that $a_1<a_2<\cdots<a_n<2^n\lambda$ and for any $k=1,2,\cdots,n$ we have \[\gcd(a_1,a_k)+\gcd(a_2,a_k)+\cdots+\gcd(a_n,a_k)\equiv 0\pmod{a_k}\]

2014 Kazakhstan National Olympiad, 1

Tags: number theory

$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$

1984 Brazil National Olympiad, 1

Tags: number theory , factorial , positive integer , factorial equation , diophantine equation

Find all solutions in positive integers to $(n+1)^k -1 = n!$

2022 Kosovo National Mathematical Olympiad, 3

Tags: number theory , prime number

Let $a,b$ and $c$ be positive integers such that $a!+b+c,b!+c+a$ and $c!+a+b$ are prime numbers. Show that $\frac{a+b+c+1}{2}$ is also a prime number.

2022 All-Russian Olympiad, 1

Tags: divisor , all russia mo , number theory

We call the $main$ $divisors$ of a composite number $n$ the two largest of its natural divisors other than $n$. Composite numbers $a$ and $b$ are such that the main divisors of $a$ and $b$ coincide. Prove that $a=b$.

2018 South East Mathematical Olympiad, 4

Tags: number theory

Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element and there exists infinitely many positive integer $m$ such that $\{m,m+2018\}\subset A$? Please prove your conclusion.

1909 Eotvos Mathematical Competition, 1

Tags: number theory , consecutive , perfect cube

Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

2016 BMT Spring, 2

Tags: number theory , algebra

Find an integer pair of solutions $(x, y)$ to the following system of equations. $$\log_2 (y^x) = 16$$ $$\log_2 (x^y) = 8$$