> 
> Determine all functions $f \colon\mathbb{R}\to\mathbb{R}$ satisfying the following two conditions:
> 
> 
> (a) $f(x +y) + f(x - y) = 2 f(x) f(y)$ for all $x, y\in\mathbb{R}$; 
> 
> 
> (b) $\lim\limits\_{x\to\infty}f(x) = 0$. 
> 
> 
> 

I found this problem in IMO 1985 longlist. I have been able to figure out that there is one solution of the form $f(x) = 0$ for all $x$.

Any other function satisfying the above has to have $f(0) = 1$ and $f(x) = f(-x)$.

However I don't know how to proceed.

Any thoughts?