Olympiad problems

2013 German National Olympiad, 2

Let $\alpha$ be a real number with $\alpha>1$. Let the sequence $(a_n)$ be defined as $$a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}$$ for all positive integers $n$. Show that there exists a positive real constant $C$ such that $a_n<C$ for all positive integers $n$.

2019 Bulgaria EGMO TST, 3

Tags: sum of digits , upper bound on sequence , ratio , number theory

In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \] as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.

2025 Romania National Olympiad, 4

Tags: sequence , upper bound on sequence , algebra

Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$. a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above. b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.