4
Part of 1950 Miklós Schweitzer
Problems(2)
Miklos Schweitzer 1950_4
Source: first round of 1950
10/2/2008
Find the polynomials having the following properties:
(i) f(0) \equal{} 1, f'(0) \equal{} f''(0) \equal{} \cdots \equal{} f^{(n)}(0) \equal{} 0
(ii) f(1) \equal{} f'(1) \equal{} f''(1) \equal{} \cdots \equal{} f^{(m)}(1) \equal{} 0
algebrapolynomialalgebra proposed
Miklos Schweitzer 1950_4
Source: second part of 1950
10/3/2008
Put
M\equal{}\begin{pmatrix}p&q&r\\
r&p&q\\q&r&p\end{pmatrix}
where and p\plus{}q\plus{}r\equal{}1. Prove that
\lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\
\frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}
linear algebramatrixlimitvectorlinear algebra unsolved