2003 AMC 10B Problem 7

Problem 7 of 25EasierAlgebra Number Theory

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}

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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.

More practice

Concepts: floor and ceiling functions · perfect square · summation

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.