12 Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$ 9 A closed form for $\int_0^\pi \lvert \sin(m t) \cos(n t) \rvert \, \mathrm{d} t$ 7 Evaluate $\lim_{n\to\infty} \prod_{k=1}^n \frac{2k}{2k-1}\int_{-1}^{\infty} \frac{{\left(\cos{x}\right)}^{2n}}{2^x} \; dx$ 7 How do I evaluate $\sum_{k = 1}^{\infty}\big[\frac{(-1)^{k - 1}}{k}\sum_{n = 0}^{\infty}\big\{\frac{1}{k(2^n) + 1}\big\}\big]$? 7 A hard inequality indian olympiad problem

### Reputation (10,505)

 +50 Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$ +35 prove inequality $\frac{\sin \pi x}{\pi x} \geq \frac{1-x^2}{1+x^2}$ for all R +20 Lower bounds for $|\sin 1| + |\sin 2| + \cdots + |\sin (3n)|$ +10 Solving recursive function with floor

### Questions (20)

 8 The limit and asymptotic analysis of $a_n^2 - n$ from $a_{n+1} = \frac{a_n}{n} + \frac{n}{a_n}$ 5 Upper bounds for $\frac{x_1}{1+x_1^2} + \frac{x_2}{1 + x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + x_2^2 + \cdots + x_n^2}$ 4 Prove $\sqrt{a + ab} + \sqrt{b} + \sqrt{c} \ge 3$ for $c = \min(a, b, c)$ and $ab + bc + ca = 2$ 4 Whether $\lim_{n\to \infty} \frac{2}{\mathsf{e}}(\sum_{k=0}^{\lfloor n/2\rfloor} \binom{n}{k}(1-\frac{2k}{n})^{n-1})^{-1/n}$ exists 4 Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

### Tags (176)

 330 inequality × 234 43 integration × 15 98 contest-math × 62 35 convex-analysis × 22 80 real-analysis × 60 35 trigonometry × 18 62 calculus × 35 34 symmetric-polynomials × 21 45 sequences-and-series × 26 32 definite-integrals × 8

### Bookmarks (137)

 263 How does one prove the determinant inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$? 128 Intuitive explanation of entropy 108 Intuition behind Conditional Expectation 104 Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$ 81 If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$