This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 20

2021 China Team Selection Test, 4
Tags: totient function , function , number theory
Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

2016 Postal Coaching, 2
Tags: totient function , number theory , inequalities
Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

2021 China Team Selection Test, 4
Tags: totient function , number theory , function
Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

1984 Balkan MO, 3
Tags: totient function , euler , number theory , modular arithmetic , function
Show that for any positive integer $m$, there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$, the representation of $5^{n}$ ends in the representation of $5^{m}$.

1992 AIME Problems, 1
Tags: totient function , number theory , prime number , search , relatively prime , function
Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms.

2020 Switzerland - Final Round, 4
Tags: euler , totient function , number theory , function
Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$

2009 Jozsef Wildt International Math Competition, W. 4
Tags: totient function , function , number theory
Let $\Phi$ denote the Euler totient function. Prove that for infinitely many $k$ we have $\Phi (2^k+1) < 2^{k-1}$ and that for infinitely many $m$ one has $\Phi (2^m+1) > 2^{m-1}$

2020 Iran MO (3rd Round), 4
Tags: number theory , totient function
Prove that for every two positive integers $a,b$ greater than $1$. there exists infinitly many $n$ such that the equation $\phi(a^n-1)=b^m-b^t$ can't hold for any positive integers $m,t$.

1983 IMO Longlists, 12
Tags: function , calculus , rotation , totient function , geometric transformation , number theory , geometry
The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?

2024 Olympic Revenge, 4
Tags: totient function , number theory
Find all positive integers $n$ such that \[2n = \varphi(n)^{\frac{2}{3}}(\varphi(n)^{\frac{2}{3}}+1)\]

2014 AIME Problems, 3
Tags: algorithm , number theory , euler , euclidean algorithm , totient function , relatively prime , function
Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and denominator have a sum of $1000$.

2018 CMIMC Number Theory, 6
Tags: euler , totient function , function , number theory
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are coprime to $n$. Find the sum of all $1<n<100$ such that $\phi(n)\mid n$.

2016 USA TSTST, 4
Tags: function , number theory , euler , totient function
Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$. Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$. [i]Proposed by Linus Hamilton[/i]

2010 Balkan MO Shortlist, N3
Tags: function , greatest common divisor , relatively prime , totient function , number theory
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2005 ITAMO, 2
Tags: number theory , modular arithmetic , function , algebra , euler , totient function
Let $h$ be a positive integer. The sequence $a_n$ is defined by $a_0 = 1$ and \[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{ if } a_n \text{ is even }\\\\a_n+h \text{ otherwise }.\end{array}\] For example, $h = 27$ yields $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$ etc. For which $h$ is there an $n > 0$ with $a_n = 1$?

PEN J Problems, 17
Tags: function , totient function , divisor function , number theory
Show that $\phi(n)+\sigma(n) \ge 2n$ for all positive integers $n$.

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.

2020 Iran MO (3rd Round), 1
Tags: number theory , divisibility , totient function
Find all positive integers $n$ such that the following holds. $$\tau(n)|2^{\sigma(n)}-1$$

2019 Durer Math Competition Finals, 3
Tags: greatest common divisor , function , relatively prime , totient function , number theory
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.

2010 Balkan MO, 4
Tags: function , greatest common divisor , number theory , relatively prime , totient function
For each integer $n$ ($n \ge 2$), let $f(n)$ denote the sum of all positive integers that are at most $n$ and not relatively prime to $n$. Prove that $f(n+p) \neq f(n)$ for each such $n$ and every prime $p$.