AMC8 1996

AMC8 1996 · Q20

AMC8 1996 · Q20. It mainly tests Functions basics, Remainders & modular arithmetic.

Suppose there is a special key on a calculator that replaces the number $x$ currently displayed with the number given by the formula $\frac{1}{1-x}$. For example, if the calculator is displaying 2 and the special key is pressed, then the calculator will display $-1$ since $\frac{1}{1-2} = -1$. Now suppose that the calculator is displaying 5. After the special key is pressed 100 times in a row, the calculator will display

假设计算器上有一个特殊按键，将当前显示的数字 $x$ 替换为公式 $\frac{1}{1-x}$ 给出的数字。例如，如果计算器显示 2，按下特殊键后，将显示 $-1$，因为 $\frac{1}{1-2} = -1$。现在假设计算器显示 5。连续按下特殊键 100 次后，计算器将显示

(A) -0.25 -0.25

(B) 0 0

(D) 1.25 1.25

(E) 5 5

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): After the special key is pressed once, the calculator display reads $-0.25$ since $1/(1-5)=1/(-4)=-0.25$. If the key is pressed again, the calculator display reads $0.8$ since $1/(1-(-0.25))=1/(1.25)=0.8$. If the key is pressed a third time, the calculator display reads $5$, since $1/(1-0.8)=1/(0.2)=5$. Thus pressing the special key three times returns to the original calculator display. The calculator display will continue to cycle through the three answer $-0.25$, $0.8$, and $5$. Since $100$ is $1$ more than a multiple of $3$, the calculator display will be $-0.25$.

答案（A）：按一次特殊按键后，计算器显示为 $-0.25$，因为 $1/(1-5)=1/(-4)=-0.25$。再按一次按键后，计算器显示为 $0.8$，因为 $1/(1-(-0.25))=1/(1.25)=0.8$。第三次按下按键时，计算器显示为 $5$，因为 $1/(1-0.8)=1/(0.2)=5$。因此，连续按三次特殊按键会回到最初的计算器显示。计算器显示将继续在 $-0.25$、$0.8$ 和 $5$ 这三个数之间循环。由于 $100$ 比 $3$ 的一个倍数多 $1$，所以计算器显示将是 $-0.25$。

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