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For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that $$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$ Let \begin{al …

I had posted the following problem on stack exchange before. Suppose $\lambda$ is a real number in $\left( 0,1\right)$, and let $n$ be a positive integer. Prove that all the roots of the polynomi …

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version. …

Let $m$ be positive integer, and consider the recursion $$x_{n+1}=\frac{1}{m+1-nx_n}.$$ Does the limit of $x_n$ exist? I'm guessing the limit doesn't exists for any $m$.

Now I found \begin{align*}\frac{S}{x^2-1}& = \sum_{k\geq 0}\frac{2k+1}{(x^{2k+1}+1)^2} - \sum_{k\geq 1} \frac{2k}{(x^{2k-1}+1)(x^{2k+1}+1)}\\ &=\sum_{k=1}^{+\infty}k\left(\dfrac{1}{x^{2k-1}+1}-\dfrac{ …

I am trying to solve this Komal problem 661: Let $K$ be a fixed positive integer. Let $(a_{0},a_{1},\cdots )$ be the sequence of real numbers that satisfies $a_{0}=-1$ and $$\sum_{i_{0},i_{1},\cdots …