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• 25 $ I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt$ is always zero for $r\in[0,1)$. Why?
• 11 If $f : [a,b]\to\Bbb R$ is continuous, are there $x_1,x_2\in (a,b)$ such that $\tfrac{f(b)-f(a)}{b-a} = \tfrac{f(x_1)-f(x_2)}{x_1-x_2}$?
• 8 How to find the limit of series? (What should I know?)
• 8 Show that $A^{-1} + B^{-1}$ is invertible when $A,B$ and $A+B$ are invertible
• 7 Given $A\in\Bbb R^{n\times n}$, is $C_A := \{SAS^{-1} : S\in GL(n,\mathbb R)\}$ connected?
• 6 $X$ is a random variable, if $\Bbb E(X^2)=1$ and $\Bbb E(X)\geq a>0$, prove that $\Bbb P(X\geq\lambda a)\geq(a-\lambda a)^2$ for $0\leq\lambda\leq 1$.
• 6 Symbol: Put a smile symbol under a plus
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