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)