39ONE OF THEM WAS OUR MATHS SIR. BUT HE DIED ON FEB7 LAST MONTH.
GIVEN a,b,c,d ≥ 0 such tha abcd=1 , prove that
\frac {1}{(1+a)^2}+\frac {1}{(1+b)^2}+\frac {1}{(1+c)^2}+\frac {1}{(1+d)^2}\geq 1
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23 Answers
62@Prophet..
I cant still believe that you did this one by Jenson's
That was the first thing I tried on this one but failed miserably...
That Is why you are TheProphet :)
341i would say the level has improved due to two reasons:
(a) the internet with sites like mathlinks and reflections.awesomemath.com etc. At mathlinks you get to see some of the best brains from all over the world at work. One of the guys who got a perfect score in this year's IMO, Alex Zhai went by the name of probability 1.01 in mathlinks. My personal favourites are torajirou (just out of school, but is so comfortable with advanced university level math), Kent Merryfield, jmerry etc.
There is a 10th class student named akashnil (passed INMO this year). he is very good.
(b) availability of ebooks-though illegal to download these are wonderful wonderful resources. I unblushingly claim to have a huge library of these :D
62Sorry Mathie. Dont over estimate CMI.
I have seen ISI bangalore. It is better than CMI at pure Mathematics.
There were 3 bhatnagar award recipients in 3 years from ISI Bangalore. (It is the highest award in Maths)
Some of them have gone to IISc now. but that is a different story.
But even that place din have any pathbreaking mathematician.
Not even 1.
It is the sad fact of life
39that has to be accepted. but CMI has got all the geniouses{at least best among india, if i am not wrong.}
but they go for fields medals rather than imo.
62Add to that.. how many Indian Journals exist?
There is one by IISc Bangalore. It sucks. (Resonance)
There are a few guys in Pune who are trying to teach for Olympiads. but they are I guess not very good either. They are good but not to the extent of the best in the world. not even close to them!
62Excellent observations Prophet.
But I think that the quality has improved over the last few years.
There were a few guys really worth mentioning,
one was Abhinav Kumar (Jamshedpur) He is the reason why that coachign called Tank minted money in Jamshedpur. He got a JEE Rank 1
He was my idol when I was in Class VIII- IX. I think he represented india at the IMO right from class VIII level and even got a couple of golds.
But yeah itsbeen more due to these students than the teachers whohave trained them!
341the romanians are nothing compared to the chinese. they are really awesome.
but, some very good mathematicians have come from eastern europe. Neumann, Erdos are some big names from Hungary. There is this teacher named Titu Andreescu whom I hugely admire. He is now in the US, and really has done tremendous service by bringing out so many problem solving books in so many topics. Somewhere else in this forum, I had observed that he had trained the dreamteam of USA in one of the IMOs and the team obtained a perfect score - all team members got full 42 marks. I wish our country had a dream teacher like that.
Unfortunately in india, we still have a copy-paste mentality. It had become a plague in another forum i visited. There teacher(s) would brazenly copy solutions from these olympiad books and claim them as their own. I would keep quiet knowing full well what was happening. But thats the sad state of affairs.
39ya sir, there INEQUALITIES ARE DEADLY. ITS BETTER TO DIE THAN TO SOLVE THEM!!!!!!!!!
sir are u a member there? if so please give profile id na
62okies :)
ro is a romanian domain
and Romanians have the reputation of killing maths olympiads if i remember corectly :)
341like i said, i stumbled upon this way in some material i downloaded like a week ago. I used to struggle a lot with this kind of problems till this method came my way.
If you guys really want to see some scary ones, see the inequalities section in mathlinks.ro. The vietnamese really go ga ga over inequalities. :D
1let a=2,b=1/2,c=3,d=1/3
so
LHS=
1/(1+2)2+1/(1+1/2)2+1/(1+3)2+1/(1+1/3)2
so
1/9+4/9+1/16+9/16
is..5/9+10/16
=5/9+5/8
=(40+45)/72
=95/72
>1.........
hence proved for one combination......similarly take for all numbers of a,b,c,d and solve.......
24itna bhi tuff nahin hai................Bhatt sir's method is innovative and easy........Jenson hi to use karna hai.........[1][1]
3arre ... kahan se app yeh naam padthe hai ........ aapko toh kisi nobel price milna chaahiye...
341No Akand dont worry. This is not for JEE at all.
Mathematician, such questions should be posted in Olympiad Corner. Otherwise it will alarm the students that is in JEE syllabus.
1oh my god, prophet u r a teacher?????
Im sorry sir (about d language) I thought u wer an aspirant too...
Im extremely sorry sir......
1wat d heck...................ill die if these kind of questions came.......
man prophet,u r smart dude
341It looked quite scary at first, but thanks to a technique i picked up recently, it resolved easily.
Let a = ex, b = ey, c = ez and d= ew
f(x) = \frac{1}{(1+e^{x})^{2}}
Then f"(x) \ge 0 (left as an exercise for the reader).
That means f is convex. Hence by Jensen's Inequality
\frac{1}{4}\sum \frac{1}{(1+e^x)^2} \ge \frac{1}{(1+e^{\frac{x+y+z+w}{4}})^2} \\ \\ = \frac{1}{(1+(abcd)^{\frac{1}{4}})^2} = \frac{1}{4}
Hence the given inequality follows