AM-GM gives abc1

 

Now a²+b²+c² = 4-abc  ab+bc+ca

 

Hence 4-2abcab+bc+ca-abc

 

4-2abc  2

 

Hence ab+bc+ca2

 

beaten to the finish

It is given that f(x), a continuous function has no real roots.

 

c = f(0) = 0

 

If f(-1) = a-b+c<0, then f(x) has a root between -1 and 0, which contradicts the fact that f has no real roots.

 

Hence a-b+c>0

 

 

1. I agree with Koni that the editor should incorporate Latex. It is useful throughout the forum. It is simple to learn and use, but more importantly, it makes eveyrone's posts more intelligible and less ambiguous.

 

2. Second, perhaps some sort of gradation like introductory, intermediate and advanced. The idea is this: I personally think this forum is a great place to hone your skills, especially if some teachers and experts participate and guide discussions. So, we should make it feasible for a new student to enter the forum without feeling intimidated because with time he/she can grow into a great contributor to the forum. Not everyone may be willing to pay the price of looking foolish for a while. That way, a participant can enter into a zone he is comfortable with and then work his way up

 

 Some really good participants pose problems that are at times above JEE level, and if I could say this not really required for JEE. Such problems could easily dent the confidence of others who are striving to get there. Placing it in 'advanced' would keep confusion out of many minds. This entails a moderator to appropriately move problems into the right categories if required.

 

3. Regarding rates: Well my fear is for some who have notched up a huge total without really exercising their mind on problems posed in the forums. Poems, wisecracks and GKNW else. I have been through some of your solutions to problems. All I can predict is that the next few months have rude surprises in store for you.

 

I possible lets restrict rates to only the problem sub-forums. Rates for articles, I dont think are an issue, yet if some feel that these are easy-come, then suggest some other incentive.

 

4. Regarding the policy on experts: I would say let market forces decide. Expert is just a badge you carry. But, the forum-members automatically gravitate to the problem solvers. artofproblemsolving.com is a good model because they have nominated only moderators and no experts.

 

 (But, if you still need experts): Being senior to most of the participants, my frank observation is that we must not have JEE aspirants as experts. You guys are young and it is easy to develop a false sort of complacence. Beware of that false pride. No one can be too good.

 

5. Smileys: Add  to Koni's list. I will find some more later, but this one is damn useful.

(a-b)² + (c-b)² = (a+c-2b)² + 2(a-b)(b-c) etc.

 

Hence  2(a+c-2b)² + 2(a-b)(b-c) = (a+c-2b)²

 

or  (a+c-2b)² + 2(a-b)(b-c) = 0

 

or 2(a-b)² = 0

 

Hence a=b=c

 

Hint: write n²+n as n(n+1).

 

Mistimed

btw, now that i have your attention kvs and rohit, how about solving that upperbound problem.

 

Looks like I have to use a ventilator to keep it alive

sorry, i made 2 mistakes:

1. I should have phrased it as there is atleast one pair of lines such that the angle between them is more than 26 deg

 

2. I should have said, pros keep out!

 

Pigeonhole principle it is.

 

Dear Admin

 

I have a suggestion for the editor.

 

 I think the number of members conversant with Latex has reached critical mass. Is it possible to incorporate Latex into the editor like in mathlinks.ro or artofproblemsolving.com. Its not very difficult to learn and use.

 

In the meanwhile dummies like me can use the texer and paste the output, but nothing like having the editor Latex-enabled

 

This way, the readability of all posts will improve considerably.

 

This will be a great help

 

Thanx