AMC10 2003 B

AMC10 2003 B · Q7

AMC10 2003 B · Q7. It mainly tests Inequalities with floors/ceilings (basic), Basic counting (rules of product/sum).

The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$. For example, $\lfloor 3 \rfloor = 3$, and $\lfloor 9/2 \rfloor = 4$. Compute $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor$.

符号 $\lfloor x \rfloor$ 表示不大于 $x$ 的最大整数。例如，$\lfloor 3 \rfloor = 3$，$\lfloor 9/2 \rfloor = 4$。计算 $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor$。

(A) 35 35

(B) 38 38

(D) 42 42

(E) 136 136

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) The first three values in the sum are 1, the next five are 2, the next seven are 3, and the final one is 4 for a total of $3\cdot 1 + 5\cdot 2 + 7\cdot 3 + 1\cdot 4 = 38.$

（B）和式中的前三个值是 1，接下来的五个是 2，再接下来的七个是 3，最后一个是 4，因此总计为 $3\cdot 1 + 5\cdot 2 + 7\cdot 3 + 1\cdot 4 = 38.$

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