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Difference between revisions of "2020 AMC 12A Problems/Problem 25"

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==Problem 25==
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==Problem==
 
The number <math>a=\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers, has the property that the sum of all real numbers <math>x</math> satisfying
 
The number <math>a=\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers, has the property that the sum of all real numbers <math>x</math> satisfying
 
<cmath> \lfloor x \rfloor \cdot \{x\} = a \cdot x^2</cmath>
 
<cmath> \lfloor x \rfloor \cdot \{x\} = a \cdot x^2</cmath>
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<math>\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332</math>
 
<math>\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332</math>
  
−
==Solution 1==
+
==Solution 1 (Solves for Floor(x))==
 +
Let <math>w=\lfloor x \rfloor</math> and <math>f=\{x\}</math> denote the whole part and the fractional part of <math>x,</math> respectively, for which <math>0\leq f<1</math> and <math>x=w+f.</math>
  
 +
We rewrite the given equation as <cmath>w\cdot f=a\cdot(w+f)^2. \hspace{38.75mm}(1)</cmath>
 +
Since <math>a\cdot(w+f)^2\geq0,</math> it follows that <math>w\cdot f\geq0,</math> from which <math>w\geq0.</math>
 +
 +
We expand and rearrange <math>(1)</math> as <cmath>af^2+(2a-1)wf+aw^2=0, \hspace{23mm}(2)</cmath> which is a quadratic with either <math>f</math> or <math>w.</math>
 +
 +
For simplicity purposes, we will treat <math>w</math> as some fixed nonnegative integer so that <math>(2)</math> is a quadratic with <math>f.</math> By the Quadratic Formula, we have <cmath>f=w\Biggl(\frac{1-2a\pm\sqrt{1-4a}}{2a}\Biggr). \hspace{25mm}(3)</cmath>
 +
If <math>w=0,</math> then <math>f=0.</math> We get <math>x=w+f=0,</math> which does not affect the sum of the solutions. Therefore, we consider the case for <math>w\geq1:</math>
 +
 +
Recall that <math>0\leq f<1,</math> so <math>\frac{1-2a\pm\sqrt{1-4a}}{2a}\geq0.</math> From the discriminant, we require that <math>0\leq1-4a<1,</math> or <cmath>0<a\leq\frac14. \hspace{54mm}(4)</cmath>
 +
 +
We consider each part of <math>0\leq f<1</math> separately:
 +
<ol style="margin-left: 1.5em;">
 +
  <li><math>f\geq0</math></li><p>
 +
From <math>(2),</math> note that <math>a>0, (2a-1)w<0,</math> and <math>aw^2>0.</math> By Descartes' Rule of Signs, we deduce that <math>(2)</math> must have two positive roots, so <math>f\geq0</math> is always valid.<p>
 +
Alternatively, from <math>(3)</math> and <math>(4),</math> note that all values of <math>a</math> for which <math>0<a\leq\frac14</math> satisfy <math>1-2a>\sqrt{1-4a}.</math> We deduce that both roots in <math>(3)</math> must be positive, so <math>f\geq0</math> is always valid.<p>
 +
  <li><math>f<1</math></li><p>
 +
We rewrite <math>(3)</math> as <cmath>f=w\Biggl(\frac{1}{2a}-1\pm\frac{\sqrt{1-4a}}{2a}\Biggr).</cmath> From <math>(4),</math> it follows that <math>\frac{1}{2a}\geq\frac{1}{1/2}=2.</math> The larger root is <cmath>f\geq w\left(2-1+2\sqrt{1-4a}\right) \geq 1\Biggl(2-1+2\sqrt{1-4\cdot\frac14}\Biggr) = 1,</cmath>
 +
which contradicts <math>f<1.</math> So, we take the smaller root, from which <cmath>f=w\Biggl(\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}\Biggr)</cmath> for some constant <math>k=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}>0.</math> We rewrite <math>f</math> as <cmath>f=wk,</cmath> in which <math>f<1</math> is valid as long as <math>k<\frac 1w.</math> Note that the solutions of <math>x</math> are generated at <cmath>w=1,2,3,\ldots,W,</cmath> up to some value <math>w=W</math> such that <math>\frac{1}{W+1}\leq k<\frac1W.</math>
 +
</ol>
 +
Now, we express <math>x</math> in terms of <math>w</math> and <math>k:</math> <cmath>x=w+f=w+wk=w(1+k).</cmath>
 +
The sum of all solutions to the original equation is <cmath>\sum_{w=1}^{W}w(1+k)=(1+k)\cdot\sum_{w=1}^{W}w=(1+k)\cdot\frac{W(W+1)}{2}=420. \hspace{10mm}(\bigstar)</cmath>
 +
As <math>1+k<1+\frac1W,</math> we conclude that <math>1+k</math> is slightly above <math>1</math> so that <math>\frac{W(W+1)}{2}</math> is slightly below <math>420,</math> or <math>W(W+1)</math> is slightly below <math>840.</math> By observations, we get <math>W=28.</math> Substituting this into <math>(\bigstar)</math> produces <math>k=\frac{1}{29},</math> which satisfies <math>\frac{1}{W+1}\leq k<\frac1W,</math> as required.
 +
 +
Finally, we solve for <math>a</math> in <math>k=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}:</math>
 +
<cmath>\begin{align*}
 +
\frac{1}{29}&=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a} \\
 +
\frac{2}{29}a&=1-2a-\sqrt{1-4a} \\
 +
\frac{60}{29}a-1&=-\sqrt{1-4a} \\
 +
\frac{60^2}{29^2}a^2-\frac{120}{29}a+1&=1-4a \\
 +
\frac{60^2}{29^2}a^2-\frac{4}{29}a&=0 \\
 +
a\left(\frac{60^2}{29^2}a-\frac{4}{29}\right)&=0.
 +
\end{align*}</cmath>
 +
Since <math>a>0,</math> we obtain <math>\frac{60^2}{29^2}a-\frac{4}{29}=0,</math> from which <cmath>a=\frac{4}{29}\cdot\frac{29^2}{60^2}=\frac{29}{900}.</cmath>
 +
The answer is <math>29+900=\boxed{\textbf{(C) } 929}.</math>
 +
 +
~MRENTHUSIASM (inspired by Math Jams's <b>2020 AMC 10/12A Discussion</b>)
 +
 +
==Solution 2 (Solves for x)==
 +
 +
Let <math>x_n</math> be a root in the interval <math>(n,n+1)</math>. In this interval, <math>\lfloor x_n \rfloor = n</math> and <math>\{x_n\}=x_n-n</math>, so we must have <math>ax_n^2 = nx_n-n^2</math>, i.e., <math>ax_n^2-nx_n+n^2=0</math>. We can homogenize this equation by setting <math>x_n=n\zeta</math>; then <math>x_1=\zeta</math>, and <math>\zeta</math> is a root of <math>a\zeta^2-\zeta+1=0</math>.
 +
 +
Suppose <math>N</math> is the largest integer for which there is such a root; we have, for <math>n=1,2,\ldots , N</math>, <cmath>n < x_n = n\zeta < n+1</cmath> Summing over <math>n\in \{1,2,\ldots , N\}</math> we get <cmath>\tfrac 12 N(N+1) < 420 = \tfrac 12 N(N+1)\zeta < \tfrac 12 N(N+3)</cmath> From the right inequality we get <math>27< N</math> and from the left one we get  <math>N<29</math>. Thus <math>N=28</math>. Using this in the middle equality we get <math>\zeta = \tfrac{30}{29}</math>. Since <math>\zeta</math> satisfies <math>a\zeta^2-\zeta+1=0</math>, we get
 +
<cmath>a = \zeta^{-2}(\zeta-1)= \tfrac{29^2}{30^2}\cdot \tfrac 1{29}= \tfrac{29}{900}.</cmath>
 +
The answer is <math>29+900=\boxed{\textbf{(C) } 929}.</math>
 +
 +
~Shihan
 +
 +
==Solution 3 (Solves for x)==
 +
First note that <math>\lfloor x\rfloor \cdot \{x\}<0</math> when <math>x<0</math> while <math>ax^2\ge 0\forall x\in \mathbb{R}</math>. Thus we only need to look at positive solutions (<math>x=0</math> doesn't affect the sum of the solutions).
 +
Next, we break <math>\lfloor x\rfloor\cdot \{x\}</math> down for each interval <math>[n,n+1)</math>, where <math>n</math> is a positive integer. Assume <math>\lfloor x\rfloor=n</math>, then <math>\{x\}=x-n</math>. This means that when <math>x\in [n,n+1)</math>, <math>\lfloor x\rfloor \cdot \{x\}=n(x-n)=nx-n^2</math>. Setting this equal to <math>ax^2</math> gives
 +
<cmath>nx-n^2=ax^2\implies ax^2-nx+n^2=0 \implies x=\frac{n\pm \sqrt{n^2-4an^2}}{2a}</cmath>
 +
We're looking at the solution with the positive <math>x</math>, which is <math>x=\frac{n-n\sqrt{1-4a}}{2a}=\frac{n}{2a}\left(1-\sqrt{1-4a}\right)</math>. Note that if <math>\lfloor x\rfloor=n</math> is the greatest <math>n</math> such that <math>\lfloor x\rfloor \cdot \{x\}=ax^2</math> has a solution, the sum of all these solutions is slightly over <math>\sum_{k=1}^{n}k=\frac{n(n+1)}{2}</math>, which is <math>406</math> when <math>n=28</math>, just under <math>420</math>. Checking this gives
 +
<cmath>\begin{align*}
 +
\sum_{k=1}^{28}\frac{k}{2a}\left(1-\sqrt{1-4a}\right)&=\frac{1-\sqrt{1-4a}}{2a}\cdot 406=420 \\
 +
\frac{1-\sqrt{1-4a}}{2a}&=\frac{420}{406}=\frac{30}{29} \\
 +
29-29\sqrt{1-4a}&=60a \\
 +
29\sqrt{1-4a}&=29-60a \\
 +
29^2-4\cdot 29^2a&=29^2+3600a^2-120\cdot 29a \\
 +
3600a^2&=116a \\
 +
a&=\frac{116}{3600}=\frac{29}{900} \implies \boxed{\textbf{(C) } 929}.
 +
\end{align*}</cmath>
 +
~ktong
 +
 +
==Remark==
 +
Let <math>f(x)=\lfloor x \rfloor \cdot \{x\}</math> and <math>g(x)=a \cdot x^2.</math>
 +
 +
We make the following table of values:
 +
 +
<cmath>\begin{array}{c|c|c|l}
 +
& & & \\ [-2ex]
 +
\boldsymbol{x} & \boldsymbol{\lfloor x \rfloor} & \boldsymbol{f(x)} & \multicolumn{1}{c}{\textbf{Equation}} \\ [1.5ex]
 +
\hline
 +
& & & \\ [-1ex]
 +
[0,1) & 0 & 0 & y=0 \\ [1.5ex]
 +
[1,2) & 1 & [0,1) & y=x-1 \\ [1.5ex]
 +
[2,3) & 2 & [0,2) & y=2x-4 \\ [1.5ex]
 +
[3,4) & 3 & [0,3) & y=3x-9 \\ [1.5ex]
 +
[4,5) & 4 & [0,4) & y=4x-16 \\ [1.5ex]
 +
\cdots & \cdots & \cdots & \cdots \\ [1.5ex]
 +
[m,m+1) & m & [0,m) & y=mx-m^2 \\ [1.5ex]
 +
\end{array}</cmath>
 +
We graph <math>f(x)</math> (in red, by branches) and <math>g(x)</math> (in blue, for <math>a=\frac{29}{900}</math>) as shown below.
 +
 +
[[File:2020 AMC 12A Problem 25.png|center]]
 +
 +
Graph in Desmos: https://www.desmos.com/calculator/ouvaiqjdzj
 +
 +
~MRENTHUSIASM
 +
 +
==Video Solution 1 (Geometry)==
 +
This video shows how things like The Pythagorean Theorem and The Law of Sines work together to solve this seemingly algebraic problem: https://www.youtube.com/watch?v=6IJ7Jxa98zw&feature=youtu.be
 +
 +
==Video Solution 2==
 +
https://www.youtube.com/watch?v=xex8TBSzKNE
 +
 +
~MathEx
 +
 +
==Video Solution 3 (by Art of Problem Solving)==
 +
https://www.youtube.com/watch?v=7_mdreGBPvg&t=428s&ab_channel=ArtofProblemSolving
 +
 +
Created by Richard Rusczyk
 +
 +
==Video Solution 4==
 +
https://youtu.be/i5b5P9RPuas
 +
 +
~MathProblemSolvingSkills
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2020|ab=A|num-b=24|after=Last Problem}}
 
{{AMC12 box|year=2020|ab=A|num-b=24|after=Last Problem}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 16:13, 21 July 2022

Problem

The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying \[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\] is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?

$\textbf{(A) } 245 \qquad \textbf{(B) } 593 \qquad \textbf{(C) } 929 \qquad \textbf{(D) } 1331 \qquad \textbf{(E) } 1332$

Solution 1 (Solves for Floor(x))

Let $w=\lfloor x \rfloor$ and $f=\{x\}$ denote the whole part and the fractional part of $x,$ respectively, for which $0\leq f<1$ and $x=w+f.$

We rewrite the given equation as \[w\cdot f=a\cdot(w+f)^2. \hspace{38.75mm}(1)\] Since $a\cdot(w+f)^2\geq0,$ it follows that $w\cdot f\geq0,$ from which $w\geq0.$

We expand and rearrange $(1)$ as \[af^2+(2a-1)wf+aw^2=0, \hspace{23mm}(2)\] which is a quadratic with either $f$ or $w.$

For simplicity purposes, we will treat $w$ as some fixed nonnegative integer so that $(2)$ is a quadratic with $f.$ By the Quadratic Formula, we have \[f=w\Biggl(\frac{1-2a\pm\sqrt{1-4a}}{2a}\Biggr). \hspace{25mm}(3)\] If $w=0,$ then $f=0.$ We get $x=w+f=0,$ which does not affect the sum of the solutions. Therefore, we consider the case for $w\geq1:$

Recall that $0\leq f<1,$ so $\frac{1-2a\pm\sqrt{1-4a}}{2a}\geq0.$ From the discriminant, we require that $0\leq1-4a<1,$ or \[0<a\leq\frac14. \hspace{54mm}(4)\]

We consider each part of $0\leq f<1$ separately:

  1. $f\geq0$

    2. From $(2),$ note that $a>0, (2a-1)w<0,$ and $aw^2>0.$ By Descartes' Rule of Signs, we deduce that $(2)$ must have two positive roots, so $f\geq0$ is always valid.

    Alternatively, from $(3)$ and $(4),$ note that all values of $a$ for which $0<a\leq\frac14$ satisfy $1-2a>\sqrt{1-4a}.$ We deduce that both roots in $(3)$ must be positive, so $f\geq0$ is always valid.

  2. $f<1$

    3. We rewrite $(3)$ as \[f=w\Biggl(\frac{1}{2a}-1\pm\frac{\sqrt{1-4a}}{2a}\Biggr).\] From $(4),$ it follows that $\frac{1}{2a}\geq\frac{1}{1/2}=2.$ The larger root is \[f\geq w\left(2-1+2\sqrt{1-4a}\right) \geq 1\Biggl(2-1+2\sqrt{1-4\cdot\frac14}\Biggr) = 1,\] which contradicts $f<1.$ So, we take the smaller root, from which \[f=w\Biggl(\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}\Biggr)\] for some constant $k=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}>0.$ We rewrite $f$ as \[f=wk,\] in which $f<1$ is valid as long as $k<\frac 1w.$ Note that the solutions of $x$ are generated at \[w=1,2,3,\ldots,W,\] up to some value $w=W$ such that $\frac{1}{W+1}\leq k<\frac1W.$

Now, we express $x$ in terms of $w$ and $k:$ \[x=w+f=w+wk=w(1+k).\] The sum of all solutions to the original equation is \[\sum_{w=1}^{W}w(1+k)=(1+k)\cdot\sum_{w=1}^{W}w=(1+k)\cdot\frac{W(W+1)}{2}=420. \hspace{10mm}(\bigstar)\] As $1+k<1+\frac1W,$ we conclude that $1+k$ is slightly above $1$ so that $\frac{W(W+1)}{2}$ is slightly below $420,$ or $W(W+1)$ is slightly below $840.$ By observations, we get $W=28.$ Substituting this into $(\bigstar)$ produces $k=\frac{1}{29},$ which satisfies $\frac{1}{W+1}\leq k<\frac1W,$ as required.

Finally, we solve for $a$ in $k=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a}:$ \begin{align*} \frac{1}{29}&=\frac{1}{2a}-1-\frac{\sqrt{1-4a}}{2a} \\ \frac{2}{29}a&=1-2a-\sqrt{1-4a} \\ \frac{60}{29}a-1&=-\sqrt{1-4a} \\ \frac{60^2}{29^2}a^2-\frac{120}{29}a+1&=1-4a \\ \frac{60^2}{29^2}a^2-\frac{4}{29}a&=0 \\ a\left(\frac{60^2}{29^2}a-\frac{4}{29}\right)&=0. \end{align*} Since $a>0,$ we obtain $\frac{60^2}{29^2}a-\frac{4}{29}=0,$ from which \[a=\frac{4}{29}\cdot\frac{29^2}{60^2}=\frac{29}{900}.\] The answer is $29+900=\boxed{\textbf{(C) } 929}.$

~MRENTHUSIASM (inspired by Math Jams's 2020 AMC 10/12A Discussion)

Solution 2 (Solves for x)

Let $x_n$ be a root in the interval $(n,n+1)$. In this interval, $\lfloor x_n \rfloor = n$ and $\{x_n\}=x_n-n$, so we must have $ax_n^2 = nx_n-n^2$, i.e., $ax_n^2-nx_n+n^2=0$. We can homogenize this equation by setting $x_n=n\zeta$; then $x_1=\zeta$, and $\zeta$ is a root of $a\zeta^2-\zeta+1=0$.

Suppose $N$ is the largest integer for which there is such a root; we have, for $n=1,2,\ldots , N$, \[n < x_n = n\zeta < n+1\] Summing over $n\in \{1,2,\ldots , N\}$ we get \[\tfrac 12 N(N+1) < 420 = \tfrac 12 N(N+1)\zeta < \tfrac 12 N(N+3)\] From the right inequality we get $27< N$ and from the left one we get $N<29$. Thus $N=28$. Using this in the middle equality we get $\zeta = \tfrac{30}{29}$. Since $\zeta$ satisfies $a\zeta^2-\zeta+1=0$, we get \[a = \zeta^{-2}(\zeta-1)= \tfrac{29^2}{30^2}\cdot \tfrac 1{29}= \tfrac{29}{900}.\] The answer is $29+900=\boxed{\textbf{(C) } 929}.$

~Shihan

Solution 3 (Solves for x)

First note that $\lfloor x\rfloor \cdot \{x\}<0$ when $x<0$ while $ax^2\ge 0\forall x\in \mathbb{R}$. Thus we only need to look at positive solutions ($x=0$ doesn't affect the sum of the solutions). Next, we break $\lfloor x\rfloor\cdot \{x\}$ down for each interval $[n,n+1)$, where $n$ is a positive integer. Assume $\lfloor x\rfloor=n$, then $\{x\}=x-n$. This means that when $x\in [n,n+1)$, $\lfloor x\rfloor \cdot \{x\}=n(x-n)=nx-n^2$. Setting this equal to $ax^2$ gives \[nx-n^2=ax^2\implies ax^2-nx+n^2=0 \implies x=\frac{n\pm \sqrt{n^2-4an^2}}{2a}\] We're looking at the solution with the positive $x$, which is $x=\frac{n-n\sqrt{1-4a}}{2a}=\frac{n}{2a}\left(1-\sqrt{1-4a}\right)$. Note that if $\lfloor x\rfloor=n$ is the greatest $n$ such that $\lfloor x\rfloor \cdot \{x\}=ax^2$ has a solution, the sum of all these solutions is slightly over $\sum_{k=1}^{n}k=\frac{n(n+1)}{2}$, which is $406$ when $n=28$, just under $420$. Checking this gives \begin{align*} \sum_{k=1}^{28}\frac{k}{2a}\left(1-\sqrt{1-4a}\right)&=\frac{1-\sqrt{1-4a}}{2a}\cdot 406=420 \\ \frac{1-\sqrt{1-4a}}{2a}&=\frac{420}{406}=\frac{30}{29} \\ 29-29\sqrt{1-4a}&=60a \\ 29\sqrt{1-4a}&=29-60a \\ 29^2-4\cdot 29^2a&=29^2+3600a^2-120\cdot 29a \\ 3600a^2&=116a \\ a&=\frac{116}{3600}=\frac{29}{900} \implies \boxed{\textbf{(C) } 929}. \end{align*} ~ktong

Remark

Let $f(x)=\lfloor x \rfloor \cdot \{x\}$ and $g(x)=a \cdot x^2.$

We make the following table of values:

\[\begin{array}{c|c|c|l} & & & \\ [-2ex] \boldsymbol{x} & \boldsymbol{\lfloor x \rfloor} & \boldsymbol{f(x)} & \multicolumn{1}{c}{\textbf{Equation}} \\ [1.5ex] \hline & & & \\ [-1ex] [0,1) & 0 & 0 & y=0 \\ [1.5ex] [1,2) & 1 & [0,1) & y=x-1 \\ [1.5ex] [2,3) & 2 & [0,2) & y=2x-4 \\ [1.5ex] [3,4) & 3 & [0,3) & y=3x-9 \\ [1.5ex] [4,5) & 4 & [0,4) & y=4x-16 \\ [1.5ex] \cdots & \cdots & \cdots & \cdots \\ [1.5ex] [m,m+1) & m & [0,m) & y=mx-m^2 \\ [1.5ex] \end{array}\] We graph $f(x)$ (in red, by branches) and $g(x)$ (in blue, for $a=\frac{29}{900}$) as shown below.

2020 AMC 12A Problem 25.png

Graph in Desmos: https://www.desmos.com/calculator/ouvaiqjdzj

~MRENTHUSIASM

Video Solution 1 (Geometry)

This video shows how things like The Pythagorean Theorem and The Law of Sines work together to solve this seemingly algebraic problem: https://www.youtube.com/watch?v=6IJ7Jxa98zw&feature=youtu.be

Video Solution 2

https://www.youtube.com/watch?v=xex8TBSzKNE

~MathEx

Video Solution 3 (by Art of Problem Solving)

https://www.youtube.com/watch?v=7_mdreGBPvg&t=428s&ab_channel=ArtofProblemSolving

Created by Richard Rusczyk

Video Solution 4

https://youtu.be/i5b5P9RPuas

~MathProblemSolvingSkills

See Also

2020 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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