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Difference between revisions of "2016 AMC 10B Problems/Problem 25"

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But among all numerators coprime to <math>n</math> so far, <math>m</math> is the largest. Therefore, choosing <math>\frac{m}{n}</math> as <math>{x}</math> increases the value <math>\lfloor n \{ x \} \rfloor</math>. Since the overall change in <math>f(x)</math> is positive as fractions <math>m/n</math> increase, we deduce that all such fractions correspond to different values of the function.  
 
But among all numerators coprime to <math>n</math> so far, <math>m</math> is the largest. Therefore, choosing <math>\frac{m}{n}</math> as <math>{x}</math> increases the value <math>\lfloor n \{ x \} \rfloor</math>. Since the overall change in <math>f(x)</math> is positive as fractions <math>m/n</math> increase, we deduce that all such fractions correspond to different values of the function.  
  
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Minor Latex Edits made by MATHWIZARD2010.
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Minor Latex Edits made by MathWizard10.
  
 
===Supplement===
 
===Supplement===
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==Solution 2==
 
==Solution 2==
  
−
<math>x = \lfloor x \rfloor + \{ x \}</math> so we have <cmath>f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor.</cmath> Clearly, the value of <math>\lfloor k \{ x \} \rfloor</math> changes only when <math>x</math> is equal to any of the fractions <math>\frac{1}{k}, \frac{2}{k} \dots \frac{k-1}{k}</math>. To get all the fractions,graphing this function gives us <math>46</math> different fractions. But on average, <math>3</math> in each of the <math>5</math> intervals don’t work. This means there are a total of <math>\fbox{\textbf{(A)}\ 32}</math> different possible values of <math>f(x)</math>.
+
<math>x = \lfloor x \rfloor + \{ x \}</math> so we have <cmath>f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor.</cmath> Clearly, the value of <math>\lfloor k \{ x \} \rfloor</math> changes only when <math>x</math> is equal to any of the fractions <math>\frac{1}{k}, \frac{2}{k} \dots \frac{k-1}{k}</math>. To get all the fractions, graphing this function gives us <math>46</math> different fractions. But on average, about one is every three fractions are repetitions of another fraction (see below). This means there are a total of <math>\fbox{\textbf{(A)}\ 32}</math> different possible values of <math>f(x)</math>.
 +
 
 +
Note: This is because all fractions with denominators <math>2\le k \le 5</math> are repetitions of another fraction with denominator <math>2k,</math> which is about <math>\frac{1}{4}</math> of all the fractions. Also, some other repeated fractions are scattered around the fractions with higher denominators. This means that we can safely estimate that about <math>\frac{1}{3}</math> of the fractions are repetitions of another fraction.
  
 
==Solution 3 (Casework)==
 
==Solution 3 (Casework)==
Line 92: Line 94:
  
 
<math>9+8+6+6+2=31</math>, <math>31+1=\fbox{\textbf{(A)}\ 32}</math>
 
<math>9+8+6+6+2=31</math>, <math>31+1=\fbox{\textbf{(A)}\ 32}</math>
 +
 +
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
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 +
==Solution 4==
 +
 +
By [https://en.wikipedia.org/wiki/Hermite%27s_identity Hermite's Identity],
 +
 +
<cmath>\begin{align*}
 +
& \lfloor kx \rfloor = \lfloor x \rfloor + \lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor\\
 +
& \lfloor kx \rfloor -k \lfloor x \rfloor = \lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor - (k-1) \lfloor x \rfloor
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\end{align*}</cmath>
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 +
Therefore,
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<cmath>\begin{align*}
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\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor) & \\
 +
&= \sum_{k=2}^{10}(\lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor - (k-1) \lfloor x \rfloor)\\
 +
&= \sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor x + \frac{i}{k} \rfloor - \lfloor x \rfloor)\\
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&= \sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor \{ x \} + \frac{i}{k} \rfloor)
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\end{align*}</cmath>
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<math>0 \le \{ x \} < 1</math>, <math>0 < \frac{j}{k}<1</math> <math>\Longrightarrow</math> <math>0 < \lfloor \{ x \} + \frac{i}{k} \rfloor < 2</math> <math>\Longrightarrow</math> <math>\lfloor \{ x \} + \frac{i}{k} \rfloor = 0 \text{ or } 1</math>
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<math>\{ x \} + \frac{i}{k} \ge 1</math> <math>\Longrightarrow</math> <math>\{ x \} \ge 1 - \frac{j}{k}</math>
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 +
Arrange <math>1 - \frac{i_j}{k_j}</math> from small to large, <math>\{ x \}</math> must fall in one interval. WLOG, suppose <math>1 - \frac{i_n}{k_n} \le \{ x \} < 1- \frac{i_{n+1}}{k_{n+1}}</math>.
 +
 +
if <math>j \le n </math>,
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<cmath>\lfloor \{ x \} + \frac{i_j}{k_j} \rfloor = 1</cmath>
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 +
if <math>j > n </math>,
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<cmath>\lfloor \{ x \} + \frac{i_j}{k_j} \rfloor = 0</cmath>
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Therefore, every distinct value of <math>\sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor \{ x \} + \frac{i}{k} \rfloor)</math> has one to one correspondence with a distinct value of <math>\frac{i}{k}</math>, <math>\frac{i}{k}</math> is not reducible, <math>(i, k) = 1</math>.
 +
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Using the [[Euler Totient Function]] as in [https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_10B_Problems/Problem_25#Supplement Solution 1's Supplement], the answer is <math>\fbox{\textbf{(A)}\ 32}</math>
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~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
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==Solution 5 (No Euler Totient Function)==
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Solution without the [[Euler Totient Function]]
 +
 +
Proceed in the same way as Solution 1 until you reach <cmath>f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor</cmath>.
 +
 +
We first count the case where all values of <math>\lfloor kx_f \rfloor</math> is 0. Now, notice that the value of <math>n/k</math> can take <math>k-1</math> values (excluding 0) since <math>n</math> must be strictly less than <math>k</math>. If we add up all <math>k-1</math> for <math>2<=k<=10</math>, we get <math>1+2+3+...+9 = 45</math>.
 +
 +
Some might be tempted to mark <math>45</math> or <math>46</math> now, but there can be repeating values. Notice that whenever the value of <math>(1/2)x_f</math> changes, the value of <math>(1/8)x_f</math> must change. This means that every case in <math>k=2</math> is covered by <math>k=8</math>. This is extended to every number that is a factor of another (3 and 9, 5 and 10). Therefore, we can eliminate <math>k=2,3,4,5</math> as they are all factors of other values of <math>k</math>, which eliminates 10 cases.
 +
 +
Within the remaining numbers, there are a couple of numbers that can still have repeating values. These are <math>(6, 9)</math>, <math>(6, 8)</math>, and <math>(8,10)</math>. The first one repeats when <math>n/k</math> is equal to <math>1/3</math> or <math>2/3</math>, eliminating 2 cases, and the second and third repeat whenever <math>n/k</math> is equal to <math>1/2</math>. This eliminates another 2 cases.
 +
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Therefore, our final answer is <math>45+1-10-2-2=32</math>, which is <math>\fbox{\textbf{(A)}\ 32}</math>.
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==Remark==
 +
 +
This problem is similar to [https://artofproblemsolving.com/wiki/index.php/1985_AIME_Problems/Problem_10 1985 AIME Problem 10]. Both problems use the [[Euler Totient Function]] to find the number of distinct values of <math>\lfloor k \{ x \} \rfloor</math>.
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]

Latest revision as of 17:23, 4 April 2023

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 1

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 $n \le 10$. Explanation for this is provided below. 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 $f(x)$.

Explanation:

Arrange all such fractions in increasing order and take a current $\frac{m}{n}$ to study. Let $p$ denote the previous fraction in the list and $x_\text{old}$ ($0 \le x_\text{old} < k$ for each $k$) be the largest so that $\frac{x_\text{old}}{k} \le p$. Since $\text{ }\text{ }\frac{m}{n} > p$, we clearly have all $x_\text{new} \ge x_\text{old}$. Therefore, the change must be nonnegative.

But among all numerators coprime to $n$ so far, $m$ is the largest. Therefore, choosing $\frac{m}{n}$ as ${x}$ increases the value $\lfloor n \{ x \} \rfloor$. Since the overall change in $f(x)$ is positive as fractions $m/n$ increase, we deduce that all such fractions correspond to different values of the function.

Minor Latex Edits made by MathWizard10.

Supplement

Here are all the distinct $\frac{m}{n}$ and $\phi(k):$

When $n=2$ , $\frac{m}{n}=\frac{1}{2}$ . $\phi(2)=1$

When $n=3$ , $\frac{m}{n}=\frac{1}{3}$ , $\frac{2}{3}$ . $\phi(3)=2$

When $n=4$ , $\frac{m}{n}=\frac{1}{4}$ , $\frac{3}{4}$ . $\phi(4)=2$

When $n=5$ , $\frac{m}{n}=\frac{1}{5}$ , $\frac{2}{5}$ , $\frac{3}{5}$ , $\frac{4}{5}$ . $\phi(5)=4$

When $n=6$ , $\frac{m}{n}=\frac{1}{6}$ , $\frac{5}{6}$ . $\phi(6)=2$

When $n=7$ , $\frac{m}{n}=\frac{1}{7}$ , $\frac{2}{7}$ , $\frac{3}{7}$ , $\frac{4}{7}$ , $\frac{5}{7}$ , $\frac{6}{7}$ . $\phi(7)=6$

When $n=8$ , $\frac{m}{n}=\frac{1}{8}$ , $\frac{3}{8}$ , $\frac{5}{8}$ , $\frac{7}{8}$ . $\phi(8)=4$

When $n=9$ , $\frac{m}{n}=\frac{1}{9}$ , $\frac{2}{9}$ , $\frac{4}{9}$ , $\frac{5}{9}$ , $\frac{7}{9}$ , $\frac{8}{9}$ . $\phi(9)=6$

When $n=10$ , $\frac{m}{n}=\frac{1}{10}$ , $\frac{3}{10}$ , $\frac{7}{10}$ , $\frac{9}{10}$ . $\phi(10)=4$

$\sum_{k=2}^{10} \phi(k)=31$

$31+1=\fbox{\textbf{(A)}\ 32}$

~isabelchen

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 average, about one is every three fractions are repetitions of another fraction (see below). This means there are a total of $\fbox{\textbf{(A)}\ 32}$ different possible values of $f(x)$.

Note: This is because all fractions with denominators $2\le k \le 5$ are repetitions of another fraction with denominator $2k,$ which is about $\frac{1}{4}$ of all the fractions. Also, some other repeated fractions are scattered around the fractions with higher denominators. This means that we can safely estimate that about $\frac{1}{3}$ of the fractions are repetitions of another fraction.

Solution 3 (Casework)

Solution $1$ is abstract. In this solution I will give a concrete explanation.

WLOG, for example, when $x$ increases from $\frac{2}{3}-\epsilon$ to $\frac{2}{3}$, $\lfloor 3 \{ x \} \rfloor$ will increase from $1$ to $2$, $\lfloor 6 \{ x \} \rfloor$ will increase from $3$ to $4$, $\lfloor 9 \{ x \} \rfloor$ will increase from $5$ to $6$. In total, $f(x)$ will increase by $3$. Because $\frac{1}{3}=\frac{2}{6}=\frac{3}{9}$, these $3$ numbers are actually $1$ distinct number to cause $f(x)$ to change. In general, when $x$ increases from $\frac{m}{n}-\epsilon$ to $\frac{m}{n}$, $\lfloor k \{ x \} \rfloor$ will increse from $k \cdot \frac{m}{n} -1$ to $k \cdot \frac{m}{n}$ if $k \cdot \frac{m}{n}$ is an integer, and the value of $f(x)$ will change. So the total number of distinct values $f(x)$ could take is equal to the number of distinct values of $\frac{m}{n}$, where $0 < \frac{m}{n}<1$ and $2 \le n \le 10$.

Solution $1$ uses Euler Totient Function to count the distinct number of $\frac{m}{n}$, I am going to use casework to count the distinct values of $\frac{m}{n}$ by not counting the duplicate ones.

When $n=10$ , $\frac{m}{n}=\frac{1}{10}$ , $\frac{2}{10}$ , $...$ , $\frac{9}{10}$ $\Longrightarrow 9$

When $n=9$ , $\frac{m}{n}=\frac{1}{9}$ , $\frac{2}{9}$ , $...$ , $\frac{8}{9}$ $\Longrightarrow 8$

When $n=8$ , $\frac{m}{n}=\frac{1}{8}$ , $\frac{2}{8}$ , $...$ , $\frac{7}{8}$ $\Longrightarrow 6$ ( $\frac{4}{8}$ is duplicate)

When $n=7$ , $\frac{m}{n}=\frac{1}{7}$ , $\frac{2}{7}$ , $...$ , $\frac{6}{7}$ $\Longrightarrow 6$

When $n=6$ , $\frac{m}{n}=\frac{1}{6}$ , $\frac{5}{6}$ $\Longrightarrow 2$ ( $\frac{2}{6}$ , $\frac{3}{6}$ , and $\frac{4}{6}$ is duplicate)

When $n=5$, $4$, $3$, $2$, all the $\frac{m}{n}$ is duplicate.

$9+8+6+6+2=31$, $31+1=\fbox{\textbf{(A)}\ 32}$

~isabelchen

Solution 4

By Hermite's Identity,

\begin{align*} & \lfloor kx \rfloor = \lfloor x \rfloor + \lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor\\ & \lfloor kx \rfloor -k \lfloor x \rfloor = \lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor - (k-1) \lfloor x \rfloor \end{align*}

Therefore, \begin{align*} \sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor) & \\ &= \sum_{k=2}^{10}(\lfloor x + \frac1k \rfloor + \lfloor x + \frac2k \rfloor + \dots + \lfloor x + \frac{k-1}{k} \rfloor - (k-1) \lfloor x \rfloor)\\ &= \sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor x + \frac{i}{k} \rfloor - \lfloor x \rfloor)\\ &= \sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor \{ x \} + \frac{i}{k} \rfloor) \end{align*}

$0 \le \{ x \} < 1$, $0 < \frac{j}{k}<1$ $\Longrightarrow$ $0 < \lfloor \{ x \} + \frac{i}{k} \rfloor < 2$ $\Longrightarrow$ $\lfloor \{ x \} + \frac{i}{k} \rfloor = 0 \text{ or } 1$

$\{ x \} + \frac{i}{k} \ge 1$ $\Longrightarrow$ $\{ x \} \ge 1 - \frac{j}{k}$

Arrange $1 - \frac{i_j}{k_j}$ from small to large, $\{ x \}$ must fall in one interval. WLOG, suppose $1 - \frac{i_n}{k_n} \le \{ x \} < 1- \frac{i_{n+1}}{k_{n+1}}$.

if $j \le n$, \[\lfloor \{ x \} + \frac{i_j}{k_j} \rfloor = 1\]

if $j > n$, \[\lfloor \{ x \} + \frac{i_j}{k_j} \rfloor = 0\]

Therefore, every distinct value of $\sum_{k=2}^{10}\sum_{i=1}^{k-1}( \lfloor \{ x \} + \frac{i}{k} \rfloor)$ has one to one correspondence with a distinct value of $\frac{i}{k}$, $\frac{i}{k}$ is not reducible, $(i, k) = 1$.

Using the Euler Totient Function as in Solution 1's Supplement, the answer is $\fbox{\textbf{(A)}\ 32}$

~isabelchen

Solution 5 (No Euler Totient Function)

Solution without the Euler Totient Function

Proceed in the same way as Solution 1 until you reach \[f(x) = \sum_{k=2}^{10} \lfloor k \{ x \} \rfloor\].

We first count the case where all values of $\lfloor kx_f \rfloor$ is 0. Now, notice that the value of $n/k$ can take $k-1$ values (excluding 0) since $n$ must be strictly less than $k$. If we add up all $k-1$ for $2<=k<=10$, we get $1+2+3+...+9 = 45$.

Some might be tempted to mark $45$ or $46$ now, but there can be repeating values. Notice that whenever the value of $(1/2)x_f$ changes, the value of $(1/8)x_f$ must change. This means that every case in $k=2$ is covered by $k=8$. This is extended to every number that is a factor of another (3 and 9, 5 and 10). Therefore, we can eliminate $k=2,3,4,5$ as they are all factors of other values of $k$, which eliminates 10 cases.

Within the remaining numbers, there are a couple of numbers that can still have repeating values. These are $(6, 9)$, $(6, 8)$, and $(8,10)$. The first one repeats when $n/k$ is equal to $1/3$ or $2/3$, eliminating 2 cases, and the second and third repeat whenever $n/k$ is equal to $1/2$. This eliminates another 2 cases.

Therefore, our final answer is $45+1-10-2-2=32$, which is $\fbox{\textbf{(A)}\ 32}$.

Remark

This problem is similar to 1985 AIME Problem 10. Both problems use the Euler Totient Function to find the number of distinct values of $\lfloor k \{ x \} \rfloor$.

~isabelchen

Video Solution

https://www.youtube.com/watch?v=zXJrdDtZNbw

See Also

2016 AMC 10B (Problems • Answer Key • Resources)
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All AMC 10 Problems and Solutions

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