Jensen's Functional Equations

Well You all may know it very well. But Its very Sexy.

it follows: f(x+y/2) = f(x)/2+f(y)/2

Now if we try to solve it.
 let us set f(0) = a (any constant)  and setting y=0
we get f(x/2)=f(x)/2+a/2

we can get : f(x+y) = f(x)+f(y) - a

so we suppose a function g(x) = f(x) - a , we get g(x+y) = g(x) +g(y)

and g(x) = cx
So f(x) = cx +a ... 

Happy New Year 2011

Well guys, Happy New Year 2011
May This Year You Rock in your life.

Best Wishes
Devansh Sharma

Functions

Well I m Very Sorry For Posting this Problem after a month or more.


This is A problem from the topic Functions. This is not very difficult and not quiet easy.

Q. Let N  Be The Set of Positive Integers. Determine All the Functions g : N ---> N  Such that
                          (g(m)+n) (m+g(n))
Is a Perfect Square for all m,n belongs to N.

This Might not Impress you But You can try it. :)

Long Run Problem !

Well This One is After a Long Time From me and I can Bet that It is One of the best.
It took all my Sleep From me ! So Do Try It.

If There is A Sequence Such that an+1  = [3an]/2. Well you Have to prove that in This sequence there can be infinite evens as well infinite odds. 

Well Try It. It is a real Good one !

Asian Pacific Olympiad

Well This problem is from APMO 2009. Just Try This and post your Solutions.

Problem.  Consider the following operation on positive real numbers written on a blackboard:
Choose a number r written on the blackboard, erase that number, and then write a
pair of positive real numbers a and b satisfying the condition 2r2 = ab on the board.
Assume that you start out with just one positive real number r on the blackboard, and
apply this operation k2 ¡ 1 times to end up with k2 positive real numbers, not necessarily
distinct. Show that there exists a number on the board which does not exceed kr.


Well Hope You will like it !!

Well This is the congruency I failed to Solve !

Well This congruency is a good thing to solve. Well it is very easy not a difficult one but still I m posting it !

p^2 is congruent to -11 modulo 36. Well you have to find all integral solution for p and you have to prove that no more solutions exist.

Something Special !

Well What do you mean by a good problem ?
It is the one whose solution should not require any prerequisites except cleverness. A high school student should not be disadvantaged compared to a professional mathematician. Yes, With problems requiring cleverness only with check the mind of the person only not by learning and knowing the whole thing.

Well I came across such a problem last night and here it is for all of you.

Problem. In a Finite sequence of real numbers, every 7-sum is negative, whereas every 11-sum is positive. Find the greatest number of terms in such a sequence. 

 Well The above was an IMO problem of 1977 and it was a problem for 6 points only. It requires nothing but a common sense generating from inside. It is not different for a 10th standard student and not for a professor.

Well Try it And do comment your responses.