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

Latest revision as of 10:10, 24 December 2024

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:

$f\geq0$

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.

$f<1$

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

Note: Using the Binomial expansion one can see that $\frac{1-\sqrt{1-4a}}{2a}$ is close to $1$ when $a$ is close to $0$ .

~tsun26

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.

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

@@ Line 17: / Line 17: @@
 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>1-4a\geq0,</math> or <math>a\leq\frac14.</math> Combining this with the precondition <math>a>0,</math> we need <cmath>0<a\leq\frac14. \hspace{54mm}(4)</cmath>
+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:
@@ Line 47: / Line 47: @@
 ==Solution 2 (Solves for x)==
-Let <math>x</math> be a root in the interval <math>(n,n+1)</math>. In this interval, <math>\lfloor x \rfloor = n</math> and <math>\{x\}=x-n</math>, so we must have <math>ax^2 = nx-n^2</math>, i.e. <cmath>x = \frac{n}{2a}\left(1 \pm \sqrt{1 -4a} \right) = \frac{2n}{1\mp \sqrt{1-4a}} </cmath>Notice that only one of the expressions gives a root in <math>(n,n+1)</math>: since <math>1- \sqrt{1-4a} < 1</math>, so <math>2n/(1- \sqrt{1-4a})>2n>n+1</math>. Thus <cmath>x  = \frac{2n}{1+ \sqrt{1-4a}} =n\zeta.</cmath>Suppose <math>N</math> is the largest integer for which there is such a root; we have <cmath>n < x = 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 = \frac{30}{29}</math>. Inverting the expression for <math>\zeta</math> to get <math>a</math>, we have <cmath>a = \frac 14 \left[1-\left(\frac{2}{\zeta}-1\right)^2\right] = \frac 14 \left[1-\left(\frac{14}{15}\right)^2\right]= \frac{29}{900}.</cmath>
+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>
@@ Line 67: / Line 71: @@
 \end{align*}</cmath>
 ~ktong
+* Note: Using the Binomial expansion one can see that <math>\frac{1-\sqrt{1-4a}}{2a}</math> is close to <math>1</math> when <math>a</math> is close to <math>0</math>.
+~tsun26
 ==Remark==
@@ Line 98: / Line 106: @@
 ==Video Solution 2==
-https://www.youtube.com/watch?v=xex8TBSzKNE ~ MathEx
+https://www.youtube.com/watch?v=xex8TBSzKNE
+~MathEx
 ==Video Solution 3 (by Art of Problem Solving)==
@@ Line 104: / Line 114: @@
 Created by Richard Rusczyk
+==Video Solution 4==
+https://youtu.be/i5b5P9RPuas
+~MathProblemSolvingSkills
 ==See Also==
 {{AMC12 box|year=2020|ab=A|num-b=24|after=Last Problem}}
 {{MAA Notice}}

2020 AMC 12A (Problems • Answer Key • Resources)
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