Olympiad problems

2010 VJIMC, Problem 1

a) Is it true that for every bijection $f:\mathbb N\to\mathbb N$ the series $$\sum_{n=1}^\infty\frac1{nf(n)}$$is convergent? b) Prove that there exists a bijection $f:\mathbb N\to\mathbb N$ such that the series $$\sum_{n=1}^\infty\frac1{n+f(n)}$$is convergent. ($\mathbb N$ is the set of all positive integers.)

2015 VJIMC, 3

[b]Problem 3[/b] Determine the set of real values of $x$ for which the following series converges, and find its sum: $$\sum_{n=1}^{\infty} \left(\sum_{\substack{k_1, k_2,\ldots , k_n \geq 0\\ 1\cdot k_1 + 2\cdot k_2+\ldots +n\cdot k_n = n}} \frac{(k_1+\ldots+k_n)!}{k_1!\cdot \ldots \cdot k_n!} x^{k_1+\ldots +k_n} \right) \ . $$

infinite sum summation real analysis convergence recurrence relations

1964 Putnam, B1

Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if $$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$ then $$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$

convergence sequences