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      /  Functional Equation
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發布者內容列
isomorphism
Not too shy to talk



註冊日: 2007-01-30
發表數: 40


 Functional Equation

I don't know the skill level of everyone on this forum, but this is a relatively easy olympiad problem for you to try:

Find all real valued functions f such that:
[f(x)+f(y)][f(u)+f(v)]=f(xu-yv)+f(xv+yu).

 2007-02-05 23:23個人資料
jerry811215
Quite a regular



註冊日: 2006-12-30
發表數: 63
Chia-yi

 Re: Functional Equation

sounds difficult , maybe i will think about it for several days . HA!!

 2007-02-06 17:44個人資料
isomorphism
Not too shy to talk



註冊日: 2007-01-30
發表數: 40


 Re: Functional Equation

What makes olympiad problems fun is that fact that you are free to explore and play around with the problem.

Usually, with functional equations, it is nice to play around, conjecture, and then prove.

 2007-02-06 19:00個人資料
isomorphism
Not too shy to talk



註冊日: 2007-01-30
發表數: 40


 Re: Functional Equation

Okay, I'll start the proof.
First of all, letting x=y=u=v=0, we get that 4f(0)^2=2f(0). This means that f(0)=0 or f(0)=1/2.

If f(0)=1/2 then letting x=y=0 shows trivially that f(x) is constant and equals 1/2 for all x.

Now see what you can derive under the condition that f(0)=0.

 2007-02-26 19:22個人資料
isomorphism
Not too shy to talk



註冊日: 2007-01-30
發表數: 40


 Re: Functional Equation

Actually I'm pretty confident this is not quite difficult. It was the first problem on a functional equations packet!

(Well, there were 2 warm-up problems beforehand.)

 2007-02-26 19:23個人資料
aa963854116
Home away from home



註冊日: 2006-01-06
發表數: 392
從未發現之89號星座

 Re: Functional Equation

Actually, I have known the answer already.
i. f(x.) = 0
ii. f(x) = 1/2
iii. f(x) = x^2

 2007-03-07 18:54個人資料
jerry811215
Quite a regular



註冊日: 2006-12-30
發表數: 63
Chia-yi

 Re: Functional Equation

by the way , I think there are many people in this website might take part in the AIME test .
So, everybody good luck!!

p.s. I think i will participate in the test in tainan , since the test won't be held in chia-yi!!

 2007-03-07 19:33個人資料
jerry811215
Quite a regular



註冊日: 2006-12-30
發表數: 63
Chia-yi

 Re: Functional Equation

to isomorphism : you will get a excellent grade on the AIME test because your math is so good. Today , I wrote a test sheet about the AIME test in 2002 . i only get fifty ! XDXD

My math still need to be improved !!!

 2007-03-07 19:37個人資料
aa963854116
Home away from home



註冊日: 2006-01-06
發表數: 392
從未發現之89號星座

 Re: Functional Equation

AIME test, it seems not easy for me.
I think I just had a great fortune on the test last year.
(ps. The questions were also easier last year.)

 2007-03-08 19:57個人資料
isomorphism
Not too shy to talk



註冊日: 2007-01-30
發表數: 40


 Re: Functional Equation

Your solution to the functional equation is correct. Solving any IMO problem is already a great feat in itself; I admire your talent in mathematics.

About the AIME, last year's test indeed was easier. I'm aiming for a 14+ this year (even though a 7 will probably get me into USAMO).

 2007-03-11 10:29個人資料


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