24only IMO questions are not dificult dear...........there are putnam ,berkeley,chinese olympiad nd russian olympiads which also have far difficult problems.....[1]
i am trying this..........will try to get in 4 days[9]
lets see who can do this. this one`s for all oldies in this site like priyam ,skygirl.eureka,srinath, prophet sir, celestine, anirudh, sankara,unique,mak, NISHANT BHAIYAN, MANISH BHAIYAN, LOKESH BHAIYAN and others. { i miss rohan and ith power}
let a,b,c≥0 satisfying ab+bc+ca=1
prove that Σ 1/√a2+bc ≥2√2
there the square root is for total denominator. ALL THE BEST.
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31 Answers
24
1@mathie.... duniya badal jaye... no compromising wid sleep... [2]
:P sry [3]
62A couple of changes to make the proof more elaborate... Only one line added and one line (which i wrote wrongly removed)
apply AM GM inequality.
equality holds when
a2 + bc = b2 + ac
thus, (a-b)(a+b)=(a-b)c so a=b or a+b=c
similarly, (a-c)(a+c)=(a-c)b so a=c or a+c=b
and (c-b)(c+b)=(c-b)a so c=b or b+c=a
assume a=b
then the three cases will reduce to a=b
and ( a=c or c=0 )
if a=b and a=c
then we have proved the result.
if a=b and c=0
then ab=1 so a=1 and b=1 and c=0 which holds bcos LHS =3 <2 root 2
now take the second case
a+b=c and a+c=b and b+c=a
cos if any other holds (i mean a=c or b=c) we will be back to our previous case.
so a+b+c=0 which is not true. hence not possible.
Is it done? Or you think there are flaws?
39tell me something, i accept it will take half hour but cant u take out half our from your sleep and do this?
its not neccesity but this question is worth it. nothing other than simple inequalities, yet a great question. thats why i siad lets see who can do this.
1If the root was not for whole denominator .............it would be better....
[7]
9a prob like this wud eat away at least 1/2 hr so u shud try this in 11th itself
9this isnt a jee prob so cant compare !!
probs like these are very interesting but time consuming
i ud advice to concentrate on jee maths for next 2 months cos it requires speed
11Sorry dude....this problem is out of my scope.....can anyone tell me how this problem compares to JEE level of toughness?
39ANIRUDH I THINK IT WOULD BE BETTER IF U CONCNTRATE ON GETTING THE SOLUTION
9heres a very stupid sol ;)
the a=b=c gives maxima ie 3√3/2
now minima by anti symmetry rule occurs when a tends to maxima with b,c tending to minima ie b,c→0 and a→1/2(0) sub and solve
39its not about getting in 4 days , its about striking the excellent way to do it.
39no sir, AS FAR AS I KNOW ITS NOT AN IMO QUESTION, IF NOT EUREKA WOULD HAVE ALREADY POINTED na .
really as far as i know its not an imo mone.
but i really loved the way i had to get to its solution. even i had to try for 4 days.
39u r right prophet sir. by the way no oldies who can do this?{excluding prophet sir, he is a math genious }
33there the square root is for total denominator. ALL THE BEST.
he wrote that...
33Σ 1/√a2+bc ≥2√2
kiska sum hai... which is variable for Σ [12]