Difference between revisions of "1972 IMO Problems/Problem 5"
(→Solution) |
(→Solution) |
||
Line 12: | Line 12: | ||
Since <math>u</math> is the least upper bound for <math>|f(x)|</math>, <math>u/|g(y)| \geq u</math>. Therefore, <math>|g(y)| \leq 1</math>. | Since <math>u</math> is the least upper bound for <math>|f(x)|</math>, <math>u/|g(y)| \geq u</math>. Therefore, <math>|g(y)| \leq 1</math>. | ||
+ | |||
+ | Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html |
Revision as of 09:44, 21 October 2014
Let and be real-valued functions defined for all real values of and , and satisfying the equation for all . Prove that if is not identically zero, and if for all , then for all .
Solution
Let be the least upper bound for for all . So, for all . Then, for all ,
Therefore, , so .
Since is the least upper bound for , . Therefore, .
Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html