Difference between revisions of "1972 IMO Problems/Problem 5"
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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>. | ||
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+ | Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html |
Revision as of 10:44, 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