T Nation

Mathematics


#1

I have been given an equation that was in the European Maths Olympiad,my professor has solved it when he was in the third year in university. But as i am one of the top 15 students in the country he gave it to me...
the equation is as follows...
f(x)+f(y)=f(x) f(a-y)+f(y) f(a-x)
and we know that f(0)=1/2

Now ,i discovered that the equation is done in three phases.
in the first and second phase we have to take once X=0 ,once y=O...after that we are left we two different equations,now to find "a" we have to put those equations in the initial one,but i get stuck because i cannot take numbers out...After that I have to find all of the functions of:
f(x) f(x)+f(y)=F(x)^2 +f(y)...
Any ideas guys?(Also I am 16, in the 10th class)


#2

just squat it


#3

Where are you stuck at?


#4

Find a janitor.


#5

I don't get it, what's the point of the question? Solve for a? Solve for the unknown functions?


#6

you haven't told us what you're looking for. i'm a math major. taken through linear alg and diff eq's but we can't help without instructions.


#7

Quit tap-dancing around the issue and solve the damn equation! True genius does not need to ask for what we are solving.

But, when in doubt, always look to Thornton Mellon: "The answer is (long pause) 3?"

DB


#8

.


#9


#10

i like your belt


#11

It's a little hard to figure out exactly what you mean in the statement of your problem, but under some reasonable assumptions, the problem doesn't seem well-posed to me.

Take the equation and substitute 0 in for both x and y.
f(0) + f(0) = f(0)f(a-0)+f(0)f(a-0)
2f(0) = f(0)f(a) + f(0)f(a)
2f(0) = 2f(0)f(a)
Since f(0) is given to be 1/2, we have
1=f(a)

Now take the equation and substitute a in for both x and y.
f(a) + f(a) = f(a)f(a-a)+f(a)f(a-a)
2f(a) = f(a)f(0)+f(a)f(0)
2f(a) = 2f(a)f(0)
Since f(a)=1, we have
1=f(0) which contradicts one of our hypotheses.


#12

umm..have you tried foam rolling that shit?


#13

Just put it in her pooper.

@tomg, you're wrong...


#14

I got stuck

step 1 let x = 0

I get f(y) = [f(a-y)-1] / 2[1-f(a)]

step 2 let x=0 y=0

f(a) = 1

When I try to plug in f(a)=1 into step 1 the world ends.


#15

It's a functional equation; this is a common type of question for IMO competitions. But what's the domain? Can we assume continuity, differentiability, etc.?


#16

what does you being 16 have to do with anything? You actually gave a whole lot of extra information..


#17

squats and milk


#18

f(x) = 1/2 for x = 0
f(x) = 1 for all other values of x

Booyah!


#19

cool


#20

This thread is reaching great levels and still has plenty of potential left.