2016 AMC 10B Problems/Problem 25

Problem

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$ , where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$ . How many distinct values does $f(x)$ assume for $x \ge 0$ ?

$\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$

Solution

Since $x = \lfloor x \rfloor + \{ x \}$ , we have

$f(x) = \sum_{k=2}^{10} (\lfloor k \lfloor x \rfloor +k \{ x \} \rfloor - k \lfloor x \rfloor)$

The function can then be simplified into

$f(x) = \sum_{k=2}^{10} ( k \lfloor x \rfloor + \lfloor k \{ x \} \rfloor - k \lfloor x \rfloor)$

which becomes

$f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor$

We can see that for each value of $k$ , $\lfloor k \{ x \} \rfloor$ can equal integers from $0$ to $k-1$ .

Clearly, the value of $\lfloor k \{ x \} \rfloor$ changes only when $x$ is equal to any of the fractions $\frac{1}{k}, \frac{2}{k} \dots \frac{k-1}{k}$ .

So we want to count how many distinct fractions less than $1$ have the form $\frac{m}{n}$ where $n \le 10$ . We can find this easily by computing

$\sum_{k=2}^{10} \phi(k)$

where $\phi(k)$ is the Euler Totient Function. Basically $\phi(k)$ counts the number of fractions with $k$ as its denominator (after simplification). This comes out to be $31$ .

Because the value of $f(x)$ is at least $0$ and can increase $31$ times, there are a total of $\fbox{\textbf{(A)}\ 32}$ different possible values of $f(x)$ .

Solution 2

$x = \lfloor x \rfloor + \{ x \}$ so we have $f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor$ .

Clearly, the value of $\lfloor k \{ x \} \rfloor$ changes only when $x$ is equal to any of the fractions $\frac{1}{k}, \frac{2}{k} \dots \frac{k-1}{k}$ .

To get all the fractions, Graphing this function gives us $46$ different fractions but on an average, $3$ in each of the $5$ intervals don’t work. This gives us $46 - 15$ = $\fbox{\textbf{(A)}\32}$ (Error compiling LaTeX. Unknown error_msg)

2016 AMC 10B (Problems • Answer Key • Resources)
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2016 AMC 10B Problems/Problem 25

Contents

Problem

Solution

Solution 2

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