MathDB

6

Part of 2004 IMC

Problems(2)

Problem 6 IMC 2004 Macedonia

Source:

7/25/2004
For every complex number zz different from 0 and 1 we define the following function f(z):=∑1log⁡4z f(z) := \sum \frac 1 { \log^4 z } where the sum is over all branches of the complex logarithm. a) Prove that there are two polynomials PP and QQ such that f(z)=P(z)Q(z)f(z) = \displaystyle \frac {P(z)}{Q(z)} for all z∈C−{0,1}z\in\mathbb{C}-\{0,1\}. b) Prove that for all z∈C−{0,1}z\in \mathbb{C}-\{0,1\} we have f(z)=z3+4z2+z6(z−1)4. f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}.
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Problem 12 IMC 2004 Macedonia

Source:

7/26/2004
For n≥0 n\geq 0 define the matrices An A_n and Bn B_n as follows: A_0 \equal{} B_0 \equal{} (1), and for every n>0 n>0 let A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & B_{n \minus{} 1} \\ \end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\ A_{n \minus{} 1} & 0 \\ \end{array} \right). Denote by S(M) S(M) the sum of all the elements of a matrix M M. Prove that S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1}), for all n,k≥2 n,k\geq 2.
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