Determine all functions $f : mathbb{N} rightarrow mathbb{N}$ such that, for every positive integer $n$, we...
Determine all functions $f : mathbb{N} rightarrow mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002,.$$
I don't know where to start as in is there a function that I can get to the solution by slightly modifying it? Any ideas
discrete-mathematics contest-math problem-solving functional-equations functional-inequalities
edited Dec 11 at 19:27
Batominovski
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asked Nov 26 at 2:28
mathnoob
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Determine all functions $f : mathbb{N} rightarrow mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002,.$$
I don't know where to start as in is there a function that I can get to the solution by slightly modifying it? Any ideas
discrete-mathematics contest-math problem-solving functional-equations functional-inequalities
edited Dec 11 at 19:27
Batominovski
33.7k33292
asked Nov 26 at 2:28
mathnoob
1,759422
migrated from cs.stackexchange.com Nov 26 at 8:26
This question came from our site for students, researchers and practitioners of computer science.
Please credit the original source.
– Apass.Jack
Nov 26 at 3:59
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Determine all functions $f : mathbb{N} rightarrow mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002,.$$
I don't know where to start as in is there a function that I can get to the solution by slightly modifying it? Any ideas
discrete-mathematics contest-math problem-solving functional-equations functional-inequalities
edited Dec 11 at 19:27
Batominovski
33.7k33292
asked Nov 26 at 2:28
mathnoob
1,759422
Determine all functions $f : mathbb{N} rightarrow mathbb{N}$ such that, for every positive integer $n$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002,.$$
I don't know where to start as in is there a function that I can get to the solution by slightly modifying it? Any ideas
discrete-mathematics contest-math problem-solving functional-equations functional-inequalities
discrete-mathematics contest-math problem-solving functional-equations functional-inequalities
edited Dec 11 at 19:27
Batominovski
33.7k33292
asked Nov 26 at 2:28
mathnoob
1,759422
edited Dec 11 at 19:27
Batominovski
33.7k33292
asked Nov 26 at 2:28
mathnoob
1,759422
edited Dec 11 at 19:27
Batominovski
33.7k33292
edited Dec 11 at 19:27
Batominovski
33.7k33292
edited Dec 11 at 19:27
Batominovski
33.7k33292
33.7k33292
asked Nov 26 at 2:28
mathnoob
1,759422
asked Nov 26 at 2:28
mathnoob
1,759422
asked Nov 26 at 2:28
mathnoob
1,759422
1,759422
migrated from cs.stackexchange.com Nov 26 at 8:26
This question came from our site for students, researchers and practitioners of computer science.
migrated from cs.stackexchange.com Nov 26 at 8:26
This question came from our site for students, researchers and practitioners of computer science.
Please credit the original source.
– Apass.Jack
Nov 26 at 3:59
add a comment |
Please credit the original source.
– Apass.Jack
Nov 26 at 3:59
Please credit the original source.
– Apass.Jack
Nov 26 at 3:59
Please credit the original source.
– Apass.Jack
Nov 26 at 3:59
add a comment |
2 Answers
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oldest
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How to start: You might start for example by checking what f(0) might be. Is f(0) = 0 possible, and if not, why not? If f(0) = 1, what can we then say about f(1)? What can we then say about f(2000) or f(2001)? Get a feeling for the problem. That's how you start.
answered Nov 26 at 8:18
gnasher729
5,9571028
add a comment |
Very old problem from the 2002 Balkan Mathematical Olympiad. You have to keep on
iterating f(f(f(...(f(n)...))) and see what inequalities you get, see e.g.
https://artofproblemsolving.com/community/c6h76767
answered Nov 26 at 8:41
Sorin Tirc
1,07511
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
How to start: You might start for example by checking what f(0) might be. Is f(0) = 0 possible, and if not, why not? If f(0) = 1, what can we then say about f(1)? What can we then say about f(2000) or f(2001)? Get a feeling for the problem. That's how you start.
answered Nov 26 at 8:18
gnasher729
5,9571028
add a comment |
How to start: You might start for example by checking what f(0) might be. Is f(0) = 0 possible, and if not, why not? If f(0) = 1, what can we then say about f(1)? What can we then say about f(2000) or f(2001)? Get a feeling for the problem. That's how you start.
answered Nov 26 at 8:18
gnasher729
5,9571028
add a comment |
How to start: You might start for example by checking what f(0) might be. Is f(0) = 0 possible, and if not, why not? If f(0) = 1, what can we then say about f(1)? What can we then say about f(2000) or f(2001)? Get a feeling for the problem. That's how you start.
answered Nov 26 at 8:18
gnasher729
5,9571028
How to start: You might start for example by checking what f(0) might be. Is f(0) = 0 possible, and if not, why not? If f(0) = 1, what can we then say about f(1)? What can we then say about f(2000) or f(2001)? Get a feeling for the problem. That's how you start.
answered Nov 26 at 8:18
gnasher729
5,9571028
answered Nov 26 at 8:18
gnasher729
5,9571028
answered Nov 26 at 8:18
gnasher729
5,9571028
answered Nov 26 at 8:18
gnasher729
5,9571028
5,9571028
add a comment |
add a comment |
Very old problem from the 2002 Balkan Mathematical Olympiad. You have to keep on
iterating f(f(f(...(f(n)...))) and see what inequalities you get, see e.g.
https://artofproblemsolving.com/community/c6h76767
answered Nov 26 at 8:41
Sorin Tirc
1,07511
add a comment |
Very old problem from the 2002 Balkan Mathematical Olympiad. You have to keep on
iterating f(f(f(...(f(n)...))) and see what inequalities you get, see e.g.
https://artofproblemsolving.com/community/c6h76767
answered Nov 26 at 8:41
Sorin Tirc
1,07511
add a comment |
Very old problem from the 2002 Balkan Mathematical Olympiad. You have to keep on
iterating f(f(f(...(f(n)...))) and see what inequalities you get, see e.g.
https://artofproblemsolving.com/community/c6h76767
answered Nov 26 at 8:41
Sorin Tirc
1,07511
Very old problem from the 2002 Balkan Mathematical Olympiad. You have to keep on
iterating f(f(f(...(f(n)...))) and see what inequalities you get, see e.g.
https://artofproblemsolving.com/community/c6h76767
answered Nov 26 at 8:41
Sorin Tirc
1,07511
answered Nov 26 at 8:41
Sorin Tirc
1,07511
answered Nov 26 at 8:41
Sorin Tirc
1,07511
answered Nov 26 at 8:41
Sorin Tirc
1,07511
1,07511
add a comment |
add a comment |
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