2003 AMC 10B Problem 7

Attempt Problem 7 of the 2003 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10B solutions, or check the answer key.

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7.

The symbolism

\lfloor x \rfloor

denotes the largest integer not exceeding

x.

For example,

\lfloor 3 \rfloor = 3,

and

\lfloor \frac{9}{2} \rfloor = 4.

Compute

\begin{gathered} \lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor \\ {}+ \cdots + \lfloor \sqrt{16} \rfloor. \end{gathered}

35

38

40

42

136

Answer: B

Concepts:floor and ceiling functions perfect square summation

Difficulty rating: 1170

Small hint:

\lfloor \sqrt{n} \rfloor = k

exactly when

k^2 \le n \lt (k+1)^2

Big hint:

Count how many of

1

through

16

give each floor value

1,

2,

3,

and

4

Solution:

The value is

1

for

n=1,2,3;

it is

2

for

n=4,\ldots,8;

it is

3

for

n=9,\ldots,15;

and it is

4

for

n=16.

The sum is

3\cdot 1 + 5\cdot 2 + 7\cdot 3 + 1\cdot 4 = 38.

Thus, the correct answer is B.