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>. | ||
− | Borrowed from http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html | + | Borrowed from [http://www.cs.cornell.edu/~asdas/imo/imo/isoln/isoln725.html] |
+ | |||
+ | == See Also == {{IMO box|year=1972|num-b=4|num-a=6}} | ||
[[Category:Olympiad Algebra Problems]] | [[Category:Olympiad Algebra Problems]] | ||
[[Category:Functional Equation Problems]] | [[Category:Functional Equation Problems]] |
Latest revision as of 15:39, 29 January 2021
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 [1]
See Also
1972 IMO (Problems) • Resources | ||
Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
All IMO Problems and Solutions |