Difference between revisions of "1981 IMO Problems/Problem 6"
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We can start by creating a list consisting of certain x an y values and their outputs. <cmath>f(0,0)=1, f(0,1)=2, f(0,2)=3, f(0,3)=4, f(0,4)=5</cmath> | We can start by creating a list consisting of certain x an y values and their outputs. <cmath>f(0,0)=1, f(0,1)=2, f(0,2)=3, f(0,3)=4, f(0,4)=5</cmath> | ||
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+ | This pattern can be proved using induction. After proving, we continue to setting a list when <math>x=2</math> <math><cmath>cat</cmath></math> |
Revision as of 21:59, 8 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.
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 | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last question |
All IMO Problems and Solutions |
https://beastacademy.com/ https://beastacademy.com/
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