Difference between revisions of "2015 AMC 12B Problems/Problem 20"
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− | Simply draw a table of values of <math>f(i,j)</math> for the first few values of <math>i</math>: | + | Simply take 30 minutes to draw a table of values of <math>f(i,j)</math> for the first few values of <math>i</math>: |
<cmath>\begin{array}{|c || c | c | c | c | c |} | <cmath>\begin{array}{|c || c | c | c | c | c |} |
Revision as of 14:21, 5 October 2019
Problem
For every positive integer , let be the remainder obtained when is divided by 5. Define a function recursively as follows:
What is ?
Solution
Simply take 30 minutes to draw a table of values of for the first few values of :
Now we claim that for , for all values . We will prove this by induction on and . The base cases for , have already been proven.
For our inductive step, we must show that for all valid values of , if for all valid values of , .
We prove this itself by induction on . For the base case, , . For the inductive step, we need if . Then, by our inductive hypothesis from our inner induction and from our outer inductive hypothesis. Thus, , completing the proof.
It is now clear that for , for all values .
Thus, .
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.