## experimentX 3 years ago this is bugging me: for any $$a, b, c, d > 0$$ prove that ${a^2 \over b} + {b^2 \over c} + {c^2 \over d} +{d^2 \over a} \geq a+ b+c+d$ Also if $$a+b+c=ab+bc+ca$$ $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a} \geq {(a + b +c)(a^2 + b^2 + c^2)\over ab + bc +ac}$

1. mukushla

man u most be on tensor and pde now...lol :)

2. experimentX

on ... just for fun. if possible quick ... man

3. experimentX

just this is bugging me so badly Exercise 1.1.8.

4. experimentX
5. experimentX

Ctrl + F ... search 1.1.8. Don't have much time to waste

6. waterineyes

Actually, this question is very easy.. @experimentX

7. experimentX

how ??

8. waterineyes

Did not you get the question clearly ?? It is saying to prove that thing..

9. experimentX

how do you prove that?

10. waterineyes

Actually, I am talking about question.. Its solution is easy or not I don't know.. Ha ha ha..

11. experimentX

already wasted 1 hrs ... can't waste another ... lol

12. waterineyes

Actually, that pdf is quite useful for me.. Thanks for giving me that..

13. experimentX

some fool upvoted me :D ... that doesn't work at all .... lol

14. mukushla

man second one's RHS will simplify to$a^2+b^2+c^2$

15. experimentX

yeah ... i know that ... just need to prove it.

16. experimentX

olypiad problems ... i find it extremely difficult even at my level man!! i need to dig a hole hide.

17. experimentX

what kind of technique is this http://www.wolframalpha.com/input/?i=Minimize [{x+%2B+2+n%2Fx%2C+n+%3C%3D+3+%26%26+x+%3E%3D+3}%2C+{x}]

18. mukushla

for the first one use AM_GM 4 times$\frac{a^2}{b}+b\ge 2a$$\frac{b^2}{c}+c\ge 2b$...

19. experimentX

crazy crafty manipulation

20. experimentX

the second one ... i can prove half of the original problem without it.

21. experimentX

the last inequality i can get bu the use of quadratic formula ... but can't find any use of ... inequality n<=3

22. experimentX

particularly I mean this http://www.wolframalpha.com/input/?i=Minimize [{x+%2B+3+n%2Fx%2C++x+%3E%3D+3}%2C+{x}]

23. experimentX

what kind of problems is this x + 3n/x ... for x >= 3

24. mukushla

sorry man i was out...how u realte this to second one?

25. experimentX

can't ...have to do without it.

26. experimentX

don't know what property is used here ..

27. mukushla

i found it from one my books i will copy it here $(a+b+c)(\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}) \geq (a+b+c)(a^2+b^2+c^2)$$\frac{ab^3}{c}+\frac{bc^3}{a}+\frac{ca^3}{b} \geq b^2c+c^2a+a^2b$use AM_GM 3 times$\frac{ab^{3}}{c}+\frac{bc^{3}}{a}\ge 2b^{2}c$...

28. experimentX

which book do you use man??

29. mukushla

oh sorry man original book is in persian and not translated to other languages :(

30. mukushla

this is 500 pages about geometry and inequalities

31. experimentX

you follow a very good writer!!

32. mukushla

yeah :)

33. mukushla

see u later santosh

34. experimentX

see ya :) thanks for help man

35. waterineyes

Are you Linkha Santosh @experimentX ?? Sorry if there is any spelling mistake in name..

36. experimentX

no mistake ... the first is my surname.

37. waterineyes

38. waterineyes

That day you said now a days, everyone is on Facebook and I was just wondering why experimentX who said those words is not on Facebook..?? Now, it is clear to me..

39. experimentX

lol ... what words?

40. waterineyes

I have written those.. "Everyone is on Facebook".. I think these you said to Yahoo that day..

41. experimentX

oh yeah ... kinda remembered on thing "if you are searched in Google instead of fB then you are successful person"

42. waterineyes

Ha ha ha... Yeah, this is true one..