Olympiad problems

2014 NZMOC Camp Selection Problems, 6

Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.

2022 Serbia National Math Olympiad, P4

Tags: inequalities , phi function , number theory

Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$, $n\in\mathbb{N}$, such that $gcd(x, n)$ is either $1$ or prime. Prove $$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$ For which $n$ does equality hold?

2006 ISI B.Math Entrance Exam, 8

Tags: number theory

Let $S$ be the set of all integers $k$, $1\leq k\leq n$, such that $\gcd(k,n)=1$. What is the arithmetic mean of the integers in $S$?

1992 Swedish Mathematical Competition, 4

Tags: inequalities , number theory , diophantine

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

2012 Finnish National High School Mathematics Competition, 5

Tags: function , number theory , induction

The [i]Collatz's function[i] is a mapping $f:\mathbb{Z}_+\to\mathbb{Z}_+$ satisfying \[ f(x)=\begin{cases} 3x+1,& \mbox{as }x\mbox{ is odd}\\ x/2, & \mbox{as }x\mbox{ is even.}\\ \end{cases} \] In addition, let us define the notation $f^1=f$ and inductively $f^{k+1}=f\circ f^k,$ or to say in another words, $f^k(x)=\underbrace{f(\ldots (f}_{k\text{ times}}(x)\ldots ).$ Prove that there is an $x\in\mathbb{Z}_+$ satisfying \[f^{40}(x)> 2012x.\]

2023 May Olympiad, 2

Tags: number theory

Let $a, b, c, d$, and $e$ be positive integers such that $a\le b\le c\le d\le e$ and that $a+b+c+d+e=1002$. a) Determine the largest possible value of $a+c+e$. b) Determine the lowest possible value of $a+c+e$.

2016 Postal Coaching, 2

Tags: number theory , prime divisor

Let $\pi (n)$ denote the largest prime divisor of $n$ for any positive integer $n > 1$. Let $q$ be an odd prime. Show that there exists a positive integer $k$ such that $$\pi \left(q^{2^k}-1\right)< \pi\left(q^{2^k}\right)<\pi \left( q^{2^k}+1\right).$$

2021 Philippine MO, 7

Tags: inequalities , number theory , algebra , philippines

Let $a, b, c,$ and $d$ be real numbers such that $a \geq b \geq c \geq d$ and $$a+b+c+d = 13$$ $$a^2+b^2+c^2+d^2=43.$$ Show that $ab \geq 3 + cd$.

1997 Miklós Schweitzer, 2

Tags: number theory , real analysis

Let A = {1,4,6, ...} be a set of natural numbers n for which n is the product of an even number of primes and n+1 is the product of an odd number of primes (taking into account the multiplicity of prime powers). Prove that the series of the reciprocals of the elements of A is divergent. In other words, $A=\{n|\lambda(n)=1$ and $\lambda(n+1)=-1\}$ , where $\lambda$ is the liouville lambda function.

2006 Kyiv Mathematical Festival, 5

Tags: number theory

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $a, b, c, d$ be positive integers and $p$ be prime number such that $a^2+b^2=p$ and $c^2+d^2$ is divisible by $p.$ Prove that there exist positive integers $e$ and $f$ such that $e^2+f^2=\frac{c^2+d^2}{p}.$

2001 India Regional Mathematical Olympiad, 7

Tags: number theory , modular arithmetic

Prove that the product of the first $1000$ positive even integers differs from the product of the first $1000$ positive odd integers by a multiple of $2001$.

2015 NZMOC Camp Selection Problems, 8

Tags: number theory , divide , divisible , divisor

Determine all positive integers $n$ which have a divisor $d$ with the property that $dn + 1$ is a divisor of $d^2 + n^2$.

2001 Moldova National Olympiad, Problem 8

Tags: number theory

Prove that every positive integer $k$ can be written as $k=\frac{mn+1}{m+n}$, where $m,n$ are positive integers.

2015 India IMO Training Camp, 2

Tags: prime , number theory

Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2001 Korea - Final Round, 1

Tags: function , number theory , modular arithmetic , quadratic , relatively prime

Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions: (i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$; (ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , divide , sum

Let $n$ be a nonnegative integer and $M =\{n^3, n^3+1, n^3+2, ..., n^3+n\}$. Consider $A$ and $B$ two nonempty, disjoint subsets of $M$ such that the sum of elements of the set $A$ divides the sum of elements of the set $B$. Prove that the number of elements of the set $A$ divides the number of elements of the set $B$.

2008 Putnam, B5

Tags: derivative , integration , greatest common divisor , function , number theory , calculus

Find all continuously differentiable functions $ f: \mathbb{R}\to\mathbb{R}$ such that for every rational number $ q,$ the number $ f(q)$ is rational and has the same denominator as $ q.$ (The denominator of a rational number $ q$ is the unique positive integer $ b$ such that $ q\equal{}a/b$ for some integer $ a$ with $ \gcd(a,b)\equal{}1.$) (Note: $ \gcd$ means greatest common divisor.)

2023 AMC 12/AHSME, 16

Tags: number theory , chicken mcnugget theorem

In Coinland, there are three types of coins, each worth $6,$ $10,$ and $15.$ What is the sum of the digits of the maximum amount of money that is impossible to have? $\textbf{(A) }11\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }10$ (I forgot the order)

Kvant 2022, M2729

Tags: number theory

Determine all positive integers $n{}$ and $m{}$ such that $m^n=n^{3m}$. [i]Proposed by I. Dorofeev[/i]

1990 Bulgaria National Olympiad, Problem 4

Tags: number theory , set

Suppose $M$ is an infinite set of natural numbers such that, whenever the sum of two natural numbers is in $M$, one of these two numbers is in $M$ as well. Prove that the elements of any finite set of natural numbers not belonging to $M$ have a common divisor greater than $1$.

2022 Argentina National Olympiad, 1

Tags: number theory

For every positive integer $n$, $P(n)$ is defined as follows: For each prime divisor $p$ of $n$ is considered the largest integer $k$ such that $p^k\le n$ and all the $p^k$ are added. For example, for $n=100=2^2 \cdot 5^2$, as $2^6<100<2^7$ and $5^2<100<5^3$, it turns out that $P(100)=2^6+5^2=89$ Prove that there are infinitely many positive integers $n$ such that $P(n)>n$..

1983 Vietnam National Olympiad, 1

Tags: modular arithmetic , greatest common divisor , number theory

Are there positive integers $a, b$ with $b \ge 2$ such that $2^a + 1$ is divisible by $2^b - 1$?

2011 Tournament of Towns, 5

Tags: number theory

Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$.

2011 AIME Problems, 1

Tags: number theory , ratio , relatively prime

Jar A contains four liters of a solution that is $45\%$ acid. Jar B contains five liters of a solution that is $48\%$ acid. Jar C contains one liter of a solution that is $k\%$ acid. From jar C, $\tfrac{m}{n}$ liters of the solution is added to jar A, and the remainder of the solution in jar C is added to jar B. At the end, both jar A and jar B contain solutions that are $50\%$ acid. Given that $m$ and $n$ are relatively prime positive integers, find $k+m+n$.

2022 Bulgaria EGMO TST, 4

Tags: shortlist , sequence , recursion , existence , admiration , number theory

Denote by $l(n)$ the largest prime divisor of $n$. Let $a_{n+1} = a_n + l(a_n)$ be a recursively defined sequence of integers with $a_1 = 2$. Determine all natural numbers $m$ such that there exists some $i \in \mathbb{N}$ with $a_i = m^2$. [i]Proposed by Nikola Velov, North Macedonia[/i]