query: Is this inequality true ? How to prove it ? $\frac{1}{2}(e^x + e^{-x}) \leq e^{\frac{x^2}{2}} $
positive_passages.docid: 30385
positive_passages.text: Proving that $\frac{e^x + e^{-x}}2 \le e^{x^2/2}$
negative_passages.docid: 38448
negative_passages.text: Show that a simple inequality holds: $a<b$ implies $\frac ab < \frac{a+1}{b+1}$
negative_passages.docid: 19896
negative_passages.text: Prove $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\geq \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$
negative_passages.docid: 137791
negative_passages.text: Proving inequality $\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}} \leq \sqrt{3 \left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$
negative_passages.docid: 145909
negative_passages.text: Prove that $\frac{1}{n}+\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{2n}\geq \frac{2}{3}$
negative_passages.docid: 124968
negative_passages.text: How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$
negative_passages.docid: 146885
negative_passages.text: How prove this inequality $\frac{a-b}{b+c}+\frac{b-c}{c+d}+\frac{c-d}{d+e}+\frac{d-e}{e+a}+\frac{e-a}{a+b}\ge 0$
negative_passages.docid: 134535
negative_passages.text: Inequality $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq a + b+ c$ when $abc = 1$
negative_passages.docid: 124654
negative_passages.text: Prove this inequality: $\frac n2 \le \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac1{2^n - 1} \le n$
negative_passages.docid: 104125
negative_passages.text: How prove this inequality??
negative_passages.docid: 107601
negative_passages.text: Use of inequality $1 - \cos (x) \leq x^2 /2$
negative_passages.docid: 34349
negative_passages.text: Prove that $\frac{1}{1+x_1}+\frac{1}{1+x_2}+\cdots+\frac{1}{1+x_n} \geq \frac{n}{\sqrt[n]{x_1x_2\cdots x_n}+1}$
negative_passages.docid: 104922
negative_passages.text: Inequality $\frac1a + \frac1b + \frac1c \leq \frac{a^8+b^8+c^8}{a^3b^3c^3}$
negative_passages.docid: 56101
negative_passages.text: Prove the inequality $\frac{a}{c+a-b}+\frac{b}{a+b-c}+\frac{c}{b+c-a}\ge{3}$
negative_passages.docid: 8207
negative_passages.text: Can I prove this inequality algebraically?
negative_passages.docid: 110790
negative_passages.text: Prove the following summation inequality: $\sum_{i=1}^n(a_i)\sum_{i=1}^n(\frac{1}{a_i})\ge n^2$
negative_passages.docid: 66707
negative_passages.text: How to prove this inequality? $ | z-1 | \le | | z | -1 | + | z| \cdot | \arg z | $