SEEMOUS 2014


Problem 1. Let {n} be a nonzero natural number and {f:\Bbb{R} \rightarrow \Bbb{R}\setminus \{0\}} be a function such that {f(2014)=1-f(2013)}. Let {x_1,..,x_n} be distinct real numbers. If

\displaystyle \left| \begin{matrix} 1+f(x_1)& f(x_2)&f(x_3) & \cdots & f(x_n) \\ f(x_1) & 1+f(x_2) & f(x_3) & \cdots & f(x_n)\\ f(x_1) & f(x_2) &1+f(x_3) & \cdots & f(x_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ f(x_1)& f(x_2) & f(x_3) & \cdots & 1+f(x_n) \end{matrix} \right|=0

prove that {f} is not continuous.

Problem 2. Consider the sequence {(x_n)} given by

\displaystyle x_1=2,\ \ x_{n+1}= \frac{x_n+1+\sqrt{x_n^2+2x_n+5}}{2},\ n \geq 2.

Prove that the sequence {y_n = \displaystyle \sum_{k=1}^n \frac{1}{x_k^2-1} ,\ n \geq 1} is convergent and find its limit.

Problem 3. Let {A \in \mathcal{M}_n (\Bbb{C})} and {a \in \Bbb{C},\ a \neq 0} such that {A-A^* =2aI_n}, where {A^* = (\overline A)^t} and {\overline A} is the conjugate matrix of {A}.

(a) Show that {|\det(A)| \geq |a|^n}.

(b) Show that if {|\det(A)|=|a|^n} then {A=aI_n}.

Problem 4. a) Prove that {\displaystyle \lim_{n \rightarrow \infty} n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx=\frac{\pi}{2}}.

b) Find the limit {\displaystyle \lim_{n \rightarrow \infty} n\left(n \int_0^n \frac{\arctan(x/n)}{x(x^2+1)}dx-\frac{\pi}{2} \right)}

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