Olympiad problems

2023 OMpD, 3

For each positive integer $x$, let $\varphi(x)$ be the number of integers $1 \leq k \leq x$ that do not have prime factors in common with $x$. Determine all positive integers $n$ such that there are distinct positive integers $a_1,a_2, \ldots, a_n$ so that the set: $$S = \{a_1, a_2, \ldots, a_n, \varphi(a_1), \varphi(a_2), \ldots, \varphi(a_n)\}$$ Have exactly $2n$ consecutive integers (in some order).

2021 South East Mathematical Olympiad, 7

Tags: euler s totient function , number theory

Determine all the pairs of positive odd integers $(a,b),$ such that $a,b>1$ and $$7\varphi^2(a)-\varphi(ab)+11\varphi^2(b)=2(a^2+b^2),$$ where $\varphi(n)$ is Euler's totient function.

2021 South East Mathematical Olympiad, 8

Tags: euler s totient function , bashing , number theory

Determine all the pairs of positive integers $(a,b),$ such that $$14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2,$$ where $\varphi(n)$ is Euler's totient function.

2021 Saudi Arabia IMO TST, 9

Tags: number theory , euler s totient function , number of divisors

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]

2011 Korea Junior Math Olympiad, 6

Tags: relatively prime , number theory , euler s totient function , coprime , divisor

For a positive integer $n$, define the set $S_n$ as $S_n =\{(a, b)|a, b \in N, lcm[a, b] = n\}$ . Let $f(n)$ be the sum of $\phi (a)\phi (b)$ for all $(a, b) \in S_n$. If a prime $p$ relatively prime to $n$ is a divisor of $f(n)$, prove that there exists a prime $q|n$ such that $p|q^2 - 1$.

2014 IFYM, Sozopol, 1

Tags: number theory , euler s totient function , divisibility

Find all pairs of natural numbers $(m,n)$, for which $m\mid 2^{\varphi(n)} +1$ and $n\mid 2^{\varphi (m)} +1$.

2020 IMO Shortlist, N6

Tags: number theory , euler s totient function , number of divisors

For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that $$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$ for all $n\ge 1$ [i]Cyprus[/i]