RB3x3x7

by math_explorer, Feb 20, 2011, 5:07 AM

solvable.
whew.

\[ \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \]
because we can pair up the terms of the LHS as follows:
\[ ((m-n+1)(m+n)) ((m-n+2)(m+n-1)) \ldots ((m)(m+1)) \]
and it's obvious that each pair does not exceed $(m^2 + m)$.

\[ 2^n n! \leq \frac{(m+n)!}{(m-n)!} \]
iff
\[ \frac{2^n}{(n+1)^{\overline{n}}} \leq \binom{m+n}{m-n} \]
but the left side doesn't exceed 1 while the RHS is a positive integer.
This post has been edited 1 time. Last edited by math_explorer, Aug 19, 2011, 9:31 AM

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