13 $\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$ 7 Sylvester's law of inertia 6 Derivative of product of matrix by vector 5 Characterization of the quasi-arithmetic mean 5 We have matrix $A\in M_{n-1\times n}(\mathbb Z)$ so that the sum of entries in each row is zero. Prove that $\det(AA^T)=nk^2.$

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 +10 Sylvester's law of inertia +30 Derivative of product of matrix by vector +10 Proving inequality: $\sum_{i=1}^n \left(a_i^7+a_i^5\right) \geq 2(\sum_{i=1}^n a_i^3)^2$ +10 $\frac{1}{(1+x)(y+z)}+ \frac{1}{(1+y)(x+z)}+ \frac{1}{(1+z)(y+x)} \geq \frac{27}{8}$

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 48 linear-algebra × 22 13 real-analysis × 5 36 matrices × 14 12 contest-math × 5 17 trigonometry × 6 11 calculus × 7 17 logarithms × 2 11 elementary-number-theory × 6 14 complex-numbers × 2 11 sequences-and-series × 5

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 799 How to study math to really understand it and have a healthy lifestyle with free time? [closed] 119 Real life applications of Topology 114 A multiplication algorithm found in a book by Paul Erdős: how does it work? 112 Am I just not smart enough? [closed] 83 If $a+b=1$ so $a^{4b^2}+b^{4a^2}\leq1$

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