Difference between revisions of "1981 IMO Problems/Problem 6"
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It follows that <math>f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3 </math> when there are 1984 2s, Q.E.D. | It follows that <math>f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3 </math> when there are 1984 2s, Q.E.D. | ||
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== Solution 2 == | == Solution 2 == | ||
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</math> where <math>a=y+3</math>. Lastly where x=4 we have <cmath>f(4,0)=13, f(4,1)=65533, f(4,2)=^5 2, f(3,3)=^6 2, f(4,4)=^7 2</cmath> where each term can be represented as <math>^a 2</math> when <math>a=y+2</math>. In <math>^a 2</math> represents tetration or 2 to the power 2 to the power 2 to the power 2 ... where <math>a</math> amount of 2s. So therefore the answer is <math>f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3 </math> with 1984 2s, 2 tetration 1983. | </math> where <math>a=y+3</math>. Lastly where x=4 we have <cmath>f(4,0)=13, f(4,1)=65533, f(4,2)=^5 2, f(3,3)=^6 2, f(4,4)=^7 2</cmath> where each term can be represented as <math>^a 2</math> when <math>a=y+2</math>. In <math>^a 2</math> represents tetration or 2 to the power 2 to the power 2 to the power 2 ... where <math>a</math> amount of 2s. So therefore the answer is <math>f(4,1981) = 2^{2\cdot ^{ . \cdot 2}} - 3 </math> with 1984 2s, 2 tetration 1983. | ||
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+ | {{IMO box|num-b=5|after=Last question|year=1981}} |
Revision as of 17:15, 12 April 2024
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.
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 with 1984 2s, 2 tetration 1983. -Multpi12
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
1981 IMO (Problems) • Resources | ||
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1 • 2 • 3 • 4 • 5 • 6 | Followed by Last question |
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