Difference between revisions of "1972 IMO Problems/Problem 5"
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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>. | ||
| − | + | ==Solution== | |
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Revision as of 10:19, 20 October 2014
Let 
and 
be real-valued functions defined for all real values of 
and 
, and satisfying the equation
![\[f(x + y) + f(x - y) = 2f(x)g(y)\]](http://latex.artofproblemsolving.com/9/7/f/97f8d013a410592034cbfed478e6a77b6573100d.png)
for all 
. Prove that if 
is not identically zero, and if 
for all , then 
for all .
