query: show this $\sum_{cyc}\frac{x}{x^2-x+1}\le\frac{8}{3}$
positive_passages.docid: 147648
positive_passages.text: Proving $\frac{a}{a^2-a+1}+\frac{b}{b^2-b+1}+\frac{c}{c^2-c+1}+\frac{d}{d^2-d+1}\le \frac{8}{3}$ given $a+b+c+d=2$
negative_passages.docid: 14563
negative_passages.text: Show $\frac{b}{c} + \frac{c}{a} + \frac{a}{b} \ge 3$ for $a,b,c > 0$.
negative_passages.docid: 145019
negative_passages.text: Prove that $\sum_{n=0}^\infty \frac{(-1)^n}{3n+1} = \frac{\pi}{3\sqrt{3}}+\frac{\log 2}{3}$
negative_passages.docid: 22524
negative_passages.text: Proving that $\frac{\pi^{3}}{32}=1-\sum_{k=1}^{\infty}\frac{2k(2k+1)\zeta(2k+2)}{4^{2k+2}}$
negative_passages.docid: 60031
negative_passages.text: Showing $\sum\limits_{j=0}^M \frac{M \choose j}{N+M \choose j} = \frac{N+M+1}{N+1}$
negative_passages.docid: 117056
negative_passages.text: How to show $\sum_{n=1}^{\infty}\frac{H_{n+1}}{n(n+1)}=2$?
negative_passages.docid: 26446
negative_passages.text: Find the sum : $\displaystyle\sum_{i=0}^n \frac{2^i}{1+x^{2^{i}}}$
negative_passages.docid: 64336
negative_passages.text: Show $\sum_{k=1}^n \frac{1}{k^2} \le 2$ and $\ln(n!) \ge 1 -n+n\ln(n)$ for all positive integers n
negative_passages.docid: 50972
negative_passages.text: Prove that $\sum_{x=1}^{n} \frac{1}{x (x+1)(x+2)} = \frac{1}{4} - \frac{1}{2 (n+1) (n+2)}$
negative_passages.docid: 136444
negative_passages.text: Evaluating $\sum_{n=0}^\infty\frac{(-1)^n}{3n+1}$
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 98305
negative_passages.text: How do I show that $\sum_{cyc} \frac {a^6}{b^2 + c^2} \ge \frac {abc(a + b + c)}2?$
negative_passages.docid: 72639
negative_passages.text: Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
negative_passages.docid: 93427
negative_passages.text: Show that $\sum_{i=1}^{n} \frac {a_i}{1+a-a_i} + \prod_{i=1}^{n} {(1-a_i)} \leq 1$
negative_passages.docid: 58633
negative_passages.text: Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
negative_passages.docid: 85626
negative_passages.text: prove $\sum_{i=1}^{n}(x_i-\bar{x})^2\lt\sum_{i=1}^{n}(x_i-a)^2$
negative_passages.docid: 69082
negative_passages.text: Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$