Olympiad problems

VMEO III 2006 Shortlist, N13

Prove the following two inequalities: 1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$ 2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$

2014 India IMO Training Camp, 1

Tags: euler totient function , number theory

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

2024 CIIM, 3

Tags: number theory , euler totient function , prime number , summation

Given a positive integer $n$, let $\phi(n)$ denote the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Find all possible positive integers $k$ for which there exist positive integers $1 \leq a_1 < a_2 < \dots < a_k$ such that: \[ \left\lfloor \frac{\phi(a_1)}{a_1} + \frac{\phi(a_2)}{a_2} + \dots + \frac{\phi(a_k)}{a_k} \right\rfloor = 2024 \]

2014 India IMO Training Camp, 1

Tags: number theory , euler totient function

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$

2014 Contests, 1

Tags: number theory , euler totient function

Let $p$ be an odd prime and $r$ an odd natural number.Show that $pr+1$ does not divide $p^p-1$