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

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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 (Comprehensive)

Let $w=\lfloor x \rfloor$ and $f=\{x\}$ denote the whole part and the fractional part of $x,$ respectively, in 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 $\begin{align*} f&=\frac{(1-2a)w\pm\sqrt{(2a-1)^2w^2-4a^2w^2}}{2a} \\ &=w\Biggl(\frac{1-2a\pm\sqrt{(2a-1)^2-4a^2}}{2a}\Biggr) \\ &=w\Biggl(\frac{1-2a\pm\sqrt{1-4a}}{2a}\Biggr). \hspace{25mm}(3) \end{align*}$

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 conclude that $1-4a\geq0,$ or $a\leq\frac14.$ Combining this with the precondition $a>0,$ we have $0<a\leq\frac14. \hspace{54mm}(4)$

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 $(3),$ solving the inequality $1-2a>\sqrt{1-4a}$ gives $0<a\leq\frac14,$ which checks $(4).$ 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 $(4),$ we deduce that $\frac{1}{2a}\geq\frac{1}{1/2}=2.$ The larger root is $\begin{align*} f&\geq w\left(2-1+2\sqrt{1-4a}\right) \\ &\geq 1\Biggl(2-1+2\sqrt{1-4\cdot\frac14}\Biggr) \\ &= 1, \\ \end{align*}$ 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 values of $x$ are generated at $w=1,2,3,\cdots,W,$ up to some value $w=W$ such that $\frac{1}{W+1}\leq k<\frac1W.$

Now, we express $x$ in terms of $w$ and $k:$ $\begin{align*} x&=w+f \\ &=w+wk \\ &=w(1+k). \end{align*}$ 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 back 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

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 breakdown $\lfloor x\rfloor\cdot \{x\}$ 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 $\sum_{k=1}^{28}\frac{k}{2a}\left(1-\sqrt{1-4a}\right)=\frac{1-\sqrt{1-4a}}{2a}\cdot 406=420$ $\frac{1-\sqrt{1-4a}}{2a}=\frac{420}{406}=\frac{30}{29}$ $29-29\sqrt{1-4a}=60a$ $29\sqrt{1-4a}=29-60a$ $29^2-4\cdot 29^2a=29^2+3600a^2-120\cdot 29a$ $3600a^2=116a$ $a=\frac{116}{3600}=\frac{29}{900} \implies \boxed{\textbf{(C) }929}$ ~ktong

Solution 3 (Condensed Version of Solution 1)

Let $1<k<2$ be the unique solution in this range. Note that $ck$ is also a solution as long as $ck < c+1$ , hence all our solutions are $k, 2k, ..., bk$ for some $b$ . This sum $420$ must be between $\frac{b(b+1)}{2}$ and $\frac{(b+1)(b+2)}{2}$ , which gives $b=28$ and $k=\frac{420}{406}=\frac{30}{29}$ . Plugging this back in gives $a=\frac{29 \cdot 1}{30^2} = \frac{29}{900} \implies \boxed{\textbf{C}}$ .

Remarks

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

We make the following table of values:

$\begin{array}{c|c|c|clc} & & & & & \\ [-1.5ex] \boldsymbol{x} & \boldsymbol{\lfloor x \rfloor} & \boldsymbol{f(x)} & & \hspace{4mm}\textbf{Equation} & \\ [1.5ex] \hline & & & & & \\ [-1ex] [0,1) & 0 & 0 & & y=0 & \\ [1.5ex] [1,2) & 1 & [0,1) & & y=x-1 & \\ [1.5ex] [2,3) & 2 & [0,2) & & y=2x-4 & \\ [1.5ex] [3,4) & 3 & [0,3) & & y=3x-9 & \\ [1.5ex] [4,5) & 4 & [0,4) & & y=4x-16 & \\ [1.5ex] \cdots & \cdots & \cdots & & \ \ \ \ \ \ \ \cdots & \\ [1.5ex] [m,m+1) & m & [0,m) & & y=mx-m^2 & \\ [2ex] \end{array}$

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

@@ Line 37: / Line 37: @@
 &= 1, \\
 \end{align*}</cmath>
-which contradicts to <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 values of <math>x</math> are generated at <cmath>w=1,2,3,\cdots,W,</cmath> up to some value <math>w=W</math> such that <math>\frac{1}{W+1}\leq k<\frac1W.</math>
+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 values of <math>x</math> are generated at <cmath>w=1,2,3,\cdots,W,</cmath> up to some value <math>w=W</math> such that <math>\frac{1}{W+1}\leq k<\frac1W.</math>
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2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Problem
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