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 13 $\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$ 6 Derivative of product of matrix by vector 6 Sylvester's law of inertia 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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 +30 Derivative of product of matrix by vector +25 Proving inequality: $\sum_{i=1}^n \left(a_i^7+a_i^5\right) \geq 2(\sum_{i=1}^n a_i^3)^2$ +55 $\frac{1}{(1+x)(y+z)}+ \frac{1}{(1+y)(x+z)}+ \frac{1}{(1+z)(y+x)} \geq \frac{27}{8}$ +10 Any “shortcuts” to proving that $\frac{\sin(x)}2+\sin^2(\frac x2)\tan(\frac x2)\to\tan(\frac x2)$

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