1972 IMO Problems/Problem 5

Let $f$ and $g$ be real-valued functions defined for all real values of $x$ and $y$, and satisfying the equation \[f(x + y) + f(x - y) = 2f(x)g(y)\] for all $x, y$. Prove that if $f(x)$ is not identically zero, and if $|f(x)| \leq 1$ for all $x$, then $|g(y)| \leq 1$ for all $y$.

Solution

Let $u>0$ be the least upper bound for $|f(x)|$ for all $x$. So, $|f(x)| \leq u$ for all $x$. Then, for all $x,y$,

$2u \geq |f(x+y)|+|f(x-y)| \geq |f(x+y)+f(x-y)| = |2f(x)g(y)|=2|f(x)||g(y)|$

Therefore, $u \geq |f(x)||g(y)|$, so $|f(x)| \leq u/|g(y)|$.

Since $u$ is the least upper bound for $|f(x)|$, $u/|g(y)| \geq u$. Therefore, $|g(y)| \leq 1$.

Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html

Invalid username
Login to AoPS