Difference between revisions of "2015 AMC 12B Problems/Problem 20"
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==Problem== | ==Problem== | ||
| + | For every positive integer <math>n</math>, let <math>\text{mod}_5 (n)</math> be the remainder obtained when <math>n</math> is divided by 5. Define a function <math>f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}</math> recursively as follows: | ||
| + | |||
| + | <cmath>f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ | ||
| + | f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ | ||
| + | f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. | ||
| + | \end{cases}</cmath> | ||
| + | |||
| + | What is <math>f(2015,2)</math>? | ||
<math>\textbf{(A)}\; 0 \qquad\textbf{(B)}\; 1 \qquad\textbf{(C)}\; 2 \qquad\textbf{(D)}\; 3 \qquad\textbf{(E)}\; 4</math> | <math>\textbf{(A)}\; 0 \qquad\textbf{(B)}\; 1 \qquad\textbf{(C)}\; 2 \qquad\textbf{(D)}\; 3 \qquad\textbf{(E)}\; 4</math> | ||
Revision as of 01:20, 7 March 2015
Problem
For every positive integer 
, let 
be the remainder obtained when is divided by 5. Define a function 
recursively as follows:
![\[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}\]](http://latex.artofproblemsolving.com/1/e/f/1efa0d34cf7ae66e852b1687a71b18515cda058b.png)
What is 
?
Solution
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. 
