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1972 IMO Problems/Problem 5: Difference between revisions

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Created page with "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 i..."
 
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Let f and g be real-valued functions defined for all real values of x and y;
Let <math>f</math> and <math>g</math> be real-valued functions defined for all real values of <math>x</math> and <math>y</math>, and satisfying the equation
and satisfying the equation
<cmath>f(x + y) + f(x - y) = 2f(x)g(y)</cmath>
for all <math>x, y</math>. Prove that if <math>f(x)</math> is not identically zero, and if <math>|f(x)| \leq 1</math> for all <math>x</math>, then <math>|g(y)| \leq 1</math> for all <math>y</math>.


      f(x+y)+f(x-y)=2f(x)g(y)
==Solution==
 
for all x, y. Prove that if f(x) is not identically zero, and if |f(x)| < or = 1 for all
x; then |g(y)| < or = 1 for all y:

Revision as of 09:19, 20 October 2014

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

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