AMC12 2015 B

AMC12 2015 B · Q20

AMC12 2015 B · Q20. It mainly tests Patterns & sequences (misc), Remainders & modular arithmetic.

For every positive integer $n$, let $\bmod_{5}(n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f:\{0,1,2,3,\dots\}\times\{0,1,2,3,4\}\to\{0,1,2,3,4\}$ recursively as follows: $$f(i,j)=\begin{cases}\bmod_{5}(j+1) & \text{if $i=0$ and $0\leq j\leq4$,}\\f(i-1,1) & \text{if $i\geq1$ and $j=0$,}\\f(i-1,f(i,j-1)) & \text{if $i\geq1$ and $1\leq j\leq4$.}\end{cases}$$ What is $f(2015,2)$?

对于每个正整数 $n$，令 $\bmod_{5}(n)$ 为 $n$ 除以5的余数。递归定义函数 $f:\{0,1,2,3,\dots\}\times\{0,1,2,3,4\}\to\{0,1,2,3,4\}$ 如下： $$f(i,j)=\begin{cases}\bmod_{5}(j+1) & \text{if $i=0$ and $0\leq j\leq4$,}\\f(i-1,1) & \text{if $i\geq1$ and $j=0$,}\\f(i-1,f(i,j-1)) & \text{if $i\geq1$ and $1\leq j\leq4$.}\end{cases}$$ $f(2015,2)$ 是多少？

(A) 0 0

(B) 1 1

(D) 3 3

(E) 4 4

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Computing from the definition leads to the following values of $f(i,j)$ for $i=0,1,2,3,4,5,6$ (the horizontal coordinate in the table) and $j=0,1,2,3,4$ (the vertical coordinate). \[ \begin{array}{c|ccccccc} 4 & 0 & 1 & 1 & 0 & 3 & 1 & 1\\ 3 & 4 & 0 & 4 & 1 & 1 & 1 & 1\\ 2 & 3 & 4 & 2 & 4 & 3 & 1 & 1\\ 1 & 2 & 3 & 0 & 3 & 1 & 1 & 1\\ 0 & 1 & 2 & 3 & 0 & 3 & 1 & 1\\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} \] It follows that $f(i,2)=1$ for all $i\ge 5$.

答案（B）：根据定义进行计算，可得到当 $i=0,1,2,3,4,5,6$（表中的水平坐标）且 $j=0,1,2,3,4$（垂直坐标）时，$f(i,j)$ 的如下取值。 \[ \begin{array}{c|ccccccc} 4 & 0 & 1 & 1 & 0 & 3 & 1 & 1\\ 3 & 4 & 0 & 4 & 1 & 1 & 1 & 1\\ 2 & 3 & 4 & 2 & 4 & 3 & 1 & 1\\ 1 & 2 & 3 & 0 & 3 & 1 & 1 & 1\\ 0 & 1 & 2 & 3 & 0 & 3 & 1 & 1\\ \hline & 0 & 1 & 2 & 3 & 4 & 5 & 6 \end{array} \] 因此，对所有 $i\ge 5$ 都有 $f(i,2)=1$。

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