AMC10 2023 B

AMC10 2023 B · Q22

AMC10 2023 B · Q22. It mainly tests Linear equations, Inequalities with floors/ceilings (basic).

How many distinct values of $x$ satisfy $\lfloor{x}\rfloor^2-3x+2=0$, where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$?

有几个不同的 $x$ 满足 $\lfloor{x}\rfloor^2-3x+2=0$，其中 $\lfloor{x}\rfloor$ 表示小于或等于 $x$ 的最大整数？

(A) \text{an infinite number} \text{an infinite number}

(B) 4 4

(D) 3 3

(E) 0 0

Answer

Correct choice: (B)

正确答案：（B）

Solution

To further grasp at this equation, we rearrange the equation into \[\lfloor{x}\rfloor^2=3x-2.\] Thus, $3x-2$ is a perfect square and nonnegative. It is now much more apparent that $x \ge 2/3,$ and that $x = 2/3$ is a solution. Additionally, by observing the RHS, $x<4,$ as \[\lfloor{4}\rfloor^2 > 3\cdot4,\] since squares grow quicker than linear functions. Now that we have narrowed down our search, we can simply test for intervals $[2/3,1], [1,2],[2,3],[3,4).$ This intuition to use intervals stems from the fact that $x=1,2$ are observable integral solutions. Notice how there is only one solution per interval, as $3x-2$ increases while $\lfloor{x}\rfloor^2$ stays the same. Finally, we see that $x=3$ does not work, however, through setting $\lfloor{x}\rfloor^2 = 9,$ $x = 11/3$ is a solution and within our domain of $[3,4).$ This provides us with solutions $\left(\frac23, 1, 2, \frac{11}{3}\right),$ thus the final answer is $\boxed{(\text{B}) \ 4}.$

为了更好地理解这个方程，我们将其重排为 $\lfloor{x}\rfloor^2=3x-2$。因此，$3x-2$ 是一个完美的平方且非负。现在很明显 $x \ge 2/3$，并且 $x = 2/3$ 是一个解。此外，通过观察右边，$x<4$，因为 $\lfloor{4}\rfloor^2 > 3\cdot4$，因为平方比线性函数增长更快。现在我们缩小了搜索范围，可以简单地在区间 $[2/3,1], [1,2],[2,3],[3,4)$ 中测试。这种使用区间的直觉来自于 $x=1,2$ 是明显的整数解的事实。注意每个区间只有一个解，因为 $3x-2$ 增加而 $\lfloor{x}\rfloor^2$ 保持不变。最后，我们看到 $x=3$ 不行，但是通过设置 $\lfloor{x}\rfloor^2 = 9$，$x = 11/3$ 是一个解且在我们的定义域 $[3,4)$ 内。这给我们解 $\left(\frac23, 1, 2, \frac{11}{3}\right)$，因此最终答案是 $\boxed{(\text{B}) \ 4}$。

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