Olympiad problems

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Found problems: 2

2021 Romania National Olympiad, 3

Let $f :\mathbb R \to\mathbb R$ a function $ n \geq 2$ times differentiable so that: $ \lim_{x \to \infty} f(x) = l \in \mathbb R$ and $ \lim_{x \to \infty} f^{(n)}(x) = 0$. Prove that: $ \lim_{x \to \infty} f^{(k)}(x) = 0 $ for all $ k \in \{1, 2, \dots, n - 1\} $, where $f^{(k)}$ is the $ k $ - th derivative of $f$.

2024 Brazil Undergrad MO, 1

Tags: natural log , prime number , sum of divisors , taylor expansion , number theory , algebra

A positive integer $n$ is called perfect if the sum of its positive divisors $\sigma(n)$ is twice $n$, that is, $\sigma(n) = 2n$. For example, $6$ is a perfect number since the sum of its positive divisors is $1 + 2 + 3 + 6 = 12$, which is twice $6$. Prove that if $n$ is a positive perfect integer, then: \[ \sum_{p|n} \frac{1}{p + 1} < \ln 2 < \sum_{p|n} \frac{1}{p - 1} \] where the sums are taken over all prime divisors $p$ of $n$.