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

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Found problems: 2

2024 Turkey Olympic Revenge, 5
Let $a$ be a positive real number. Prove that a) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} > a$. b) There exists $n\in \mathbb{N}$ with $\frac{\sigma(\varphi(n))}{\varphi(\sigma(n))} < a$. (As usual, $\sigma(n) = \sum_{d\mid n} d$ and $\varphi(n)$ is the number of integers $1\le m\le n$ which are coprime with $n$.) Proposed by [i]Deniz Can Karaçelebi[/i]
phi function number theory sigma function algebra modular arithmetic

2024 Brazil Cono Sur TST, 3
Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]
divisor tau function ceiling function number theory sigma function bijection floor and ceiling functions