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ThePerfectHacker · #1 June 10th, 2007, 09:56 PM

This one is not so bad.

1)Let $f(x)$ be an $n$ -th degree* polynomial function such that $f(x)\geq 0$ . Define the function $g(x)$ as:
$g(x) = f(x) + f'(x) + f''(x) + ... + f^{(n)}(x)$ . Show that $g(x)\geq 0$ .

2)What is the least number of moves that a player can make to give a checkmate?

*)And the condition that $f(x) \not \equiv 0$ because the degree of a zero polynomial is not defined. The degree of a constant non-zero polynomial is defined to be zero. However, some authors in field theory differ on their defintions of the degree of the zero polynomial. Some define it to be $-1$ and other to be $\infty$ . The way I learned it the zero polynomial had an undefined degree. This is why I make such a comment just in case you spotted the mistake in my first sentence.