1981 IMO Problems/Problem 6
Problem
The function satisfies
(1)
(2)
(3)
for all non-negative integers . Determine
.
Solution
We observe that and that
, so by induction,
. Similarly,
and
, yielding
.
We continue with ;
;
; and
;
.
It follows that when there are 1984 2s, Q.E.D.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
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Solution 2
We can start by creating a list consisting of certain x an y values and their outputs.
This pattern can be proved using induction. After proving, we continue to setting a list when .
This pattern can also be proved using induction. The pattern seems d up of a common difference of 1. Moving on to
All of the numbers are being expressed in the form of
where
. Lastly where x=4 we have
where each term can be represented as
when
. In
represents tetration or 2 to the power 2 to the power 2 to the power 2 ... where
amount of 2s. So therefore the answer is $$ (Error compiling LaTeX. Unknown error_msg)f(3,1981)=
2$$ (Error compiling LaTeX. Unknown error_msg)