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)
[quote]Marlind wrote:
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)
[/quote]
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.
[quote]JLu wrote:
I don’t get it, what’s the point of the question? Solve for a? Solve for the unknown functions?[/quote]
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.?
