Strange \(a,b,c \in \mathbb{N}^+\) and \(41/42\)

05032010, 05:41 PM
(This post was last modified: 05032010 05:43 PM by elim.)
Post: #1




Strange \(a,b,c \in \mathbb{N}^+\) and \(41/42\)
If \(a,b,c \in \mathbb{N}^+,\quad \displaystyle{\frac{1}{a}+\frac{1}{b}+\frac{1}{c} < 1}\), prove that \(\displaystyle{\frac{1}{a}+\frac{1}{b}+\frac{1}{c} < \frac{41}{42}} \)


05072010, 10:59 AM
(This post was last modified: 05072010 10:59 AM by elim.)
Post: #2




RE: Strange \(a,b,c \in \mathbb{N}^+\) and \(41/42\)
Assume that \(a \le b \le c\), with the restriction \(s = 1/a+1/b+1/c <1\), we see that
when \(a \ge 4\), \(s \le 1/4+1/4+1/4 = 3/4 <41/42<1\) when \(a = 3\), \(s \le 1/3+1/3+1/4 < 41/42\) when \(a = 2\), \(s \le max(1/2+1/3+1/7,1/2+1/4+1/5)=41/42\) 

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