Difference between revisions of "1990 OIM Problems/Problem 1"
(Created page with "== Problem == Let <math>f</math> be a function defined in the set of integers greater or equal to zero such that: (i) If <math>n=2^j-1</math>, for all <math>n=0, 1, 2, \cdots...") |
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<cmath>f(n)+n=2^k-1</cmath> | <cmath>f(n)+n=2^k-1</cmath> | ||
| − | b. Calculate <math>f(2^1990)</math>. | + | b. Calculate <math>f(2^{1990})</math>. |
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ~translated into English by Tomas Diaz. ~orders@tomasdiaz.com | ||
Revision as of 23:51, 22 December 2023
Problem
Let 
be a function defined in the set of integers greater or equal to zero such that:
(i) If 
, for all 
then 
(ii) If 
, for all then 
a. Prove that for all integer 
, greater or equal to zero, there exist an integer 
grater than zero such that
![\[f(n)+n=2^k-1\]](http://latex.artofproblemsolving.com/e/3/4/e343172b4175bc61591e8fc2ab0e4a43b4d36323.png)
b. Calculate 
.
~translated into English by Tomas Diaz. ~orders@tomasdiaz.com
Solution
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