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Found problems: 2

2020 Kyiv Mathematical Festival, 1.2
Tags: sum , combinatorial sum , algebra
Prove that (a) for each $n \ge 1$ $$\sum_{k=0}^n C_{n}^{k} \left(\frac{k}{n}-\frac{1}{2} \right)^2 \frac{1}{2^n}=\frac{1}{4n}$$ (b) for every n \ge m \ge 2 $$\sum_{\ell=0}^n \sum_{k_1+...+k_n=\ell,k_i=0,...,m} \frac{\ell!}{k_1!...k_n!} \frac{1}{(m+1)^n} \left(\frac{\ell}{n}-\frac{m}{2} \right)^2= \left(\frac{m^3-3m^2}{12(m+1)}+\frac{m}{2}-\frac{m}{3(m+1)}\right)n$$

2015 ELMO Problems, 2
Tags: combinatorial sum , number theory
Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \] [i]Proposed by Yang Liu[/i]