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| Math Central | Quandaries & Queries |
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Question from ABOU, a teacher: good morning.......a b c are real positive no zero......proof that sq root(2a/(a+b))+sq root(2b/(b+c))+sq root(2c/(c+a))inferior or equal 3 thank you |
Hi Abou,
For the sake of the proof, let's say a≤b≤c. Then we know
2a≤2b≤2c
a+b≤2b≤b+c
2a≤a+b≤c+a
a+c≤b+c≤2c
2a≤a+b and 2b≤b+c implies that
2a/(a+b)≤1
2b/(b+c)≤1
a+c ≤ 2c implies that
1≤ 2c/a+c
But if we decrease the denominator then the proportion becomes larger so
1≤ 2c/(a+c) ≤ 2c/c = 2
So we know that √[2a/(a+b)] ≤ 1, √[2b/(b+c)] ≤ 1 and √[2c/(a+c)] ≤ √2
Hope this helps,
Janice
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