Olympiad problems

2016 VJIMC, 4

Find the value of sum $\sum_{n=1}^\infty A_n$, where $$A_n=\sum_{k_1=1}^\infty\cdots\sum_{k_n=1}^\infty \frac{1}{k_1^2}\frac{1}{k_1^2+k_2^2}\cdots\frac{1}{k_1^2+\cdots+k_n^2}.$$

2022 VTRMC, 4

Tags: logarithm , series , summation , infinite sum

Calculate the exact value of the series $\sum _{n=2} ^\infty \log (n^3 +1) - \log (n^3 - 1)$ and provide justification.

2015 VJIMC, 3

Tags: real analysis , summation , convergence , infinite sum

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

2016 Mathematical Talent Reward Programme, MCQ: P 2

Tags: function , limit , summation , infinite sum

Let $f$ be a function satisfying $f(x+y+z)=f(x)+f(y)+f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1)=1$, $f(2)=2$. Then $\lim \limits_{n\to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals [list=1] [*] 4 [*] 6 [*] 12 [*] 24 [/list]