IMO Shortlist 1972 problem 1
Dodao/la:
arhiva2. travnja 2012. ![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
are real-valued functions defined on the real line. For all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y, f(x+y)+f(x-y)=2f(x)g(y)](/media/m/7/e/7/7e783f27baead468ee8a3c9f05229672.png)
.
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
is not identically zero and
![|f(x)|\le1](/media/m/c/4/1/c41a7458631f53c450b3e2648e29cb32.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
. Prove that
![|g(x)|\le1](/media/m/9/0/d/90d1a33bb9d969af5d57f42e25a9c2eb.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
.
%V0
$f$ and $g$ are real-valued functions defined on the real line. For all $x$ and $y, f(x+y)+f(x-y)=2f(x)g(y)$. $f$ is not identically zero and $|f(x)|\le1$ for all $x$. Prove that $|g(x)|\le1$ for all $x$.
Izvor: Međunarodna matematička olimpijada, shortlist 1972