Expert: Sherman D. - 1/29/2006

Question
Prove the following if a,b,c,d >0:
1a) a/b + b/c + c/a >or= 3
b) a/b + b/c + c/d + d/a >or= 4

thank you

Answer
Lets just use 1,2,3,4
a = 1
b = 2
c = 3
d = 4

1.)
a.)
(a/b) + (b/c) + (c/a) >= 3
Multiply everything by abc
a^2c + ab^2 + bc^2 >= 3abc
(1 * 3) + (1 * 2^2) + (2 * 3^2) >= 3(1)(2)(3)
3 + 4 + 18 >= 3(6)
25 >= 18

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b.)
(a/b) + (b/c) + (c/d) + (d/a) >= 4
Multiply everything by abcd
a^2cd + ab^2d + abc^2 + bcd^2 >= 4abcd
(1*3*4) + (1*2^2*4) + (1*2*3^2) + (2*3*4^2) >= 4(1*2*3*4)
12 + 16 + 18 + 96 >= 96

Also as you can see, if at anytime a, b, c, or d is 0, the problem becomes undefined. And if you change one of them to a negative, then you would have to change another problem to a negative.

I hope this helps, if not, then sorry, i am not to good a some proofs.