View Full Version : Bilkent University Problem of the Month
jay_shark
02-13-2007, 11:54 AM
The following problem is this months problem of the month . If you know the solution to this problem you can get credit for it on their website . http://www.fen.bilkent.edu.tr/~cvmath/Problem/0702q.pdf
jay_shark
02-13-2007, 03:44 PM
For two positive reals a1,a2 , is this true? if so , then perhaps there is a mathematical induction argument that works .
[1+1/(a1^2+a1^4)][1+1/a2^2+a2^4] >=[1+1/(a1a2+(a1a2)^2)]^2
hmmm...
I'm not sure if 1024 plays an important part in this inequality but I doubt it .
Enrique
02-13-2007, 04:22 PM
[ QUOTE ]
For two positive reals a1,a2 , is this true? if so , then perhaps there is a mathematical induction argument that works .
[1+1/(a1^2+a1^4)][1+1/a2^2+a2^4] >=[1+1/(a1a2+(a1a2)^2)]^2
hmmm...
I'm not sure if 1024 plays an important part in this inequality but I doubt it .
[/ QUOTE ]
I tried that too.
I tried using repeated AM-GM and it didn't work. I have stopped trying the problem for about a week, maybe I should give it a shot again.
jay_shark
02-13-2007, 04:26 PM
I just looked at it today but so far no luck . Maybe some caffeine will inspire me .
coffee time .
jay_shark
02-13-2007, 05:05 PM
I verified that it's true for two numbers .I'll show the details later when I hopefully give out a solution .
For two numbers , it turns out that the numerator on the lhs is always >= to the numerator on the rhs . Also , the denominator on the lhs is always <= than the denominator on the rhs . This shows that the lhs is always >= to the rhs .
You have to expand the expression to see that this is true for two numbers . Then mathematical induction should do the trick .
jay_shark
02-13-2007, 05:08 PM
Also , i'm very certain that this works only for powers of 2 . Notice that 1024 is 2^10 .
Now induction should certainly be a lot easier to deal with .
thylacine
02-13-2007, 07:57 PM
Suggestion:
Let f(x)=1+1/(x+x^2)
Show that for all positive real a,b
f(a^2)f(b^2) >= [f(ab)]^2
or look at concavity of g(z) = f(e^z)
(I have not checked what I am saying)
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