Difference between revisions of "2021 JMPSC Sprint Problems/Problem 1"

Latest revision as of 09:53, 12 July 2021

Problem

Compute $\left(\frac{1}{3}+\frac{1}{6}+\frac{1}{9}\right)(3+6+9)$ .

Solution

Solving the right side gives $3 + 6 + 9 = 18$ . Distributing into the left side gives $\frac{18}{3}+\frac{18}{6}+\frac{18}{9}$ , so the answer is $6 + 3 + 2 = \boxed{11}$ .

Solution 2

$\frac{1}{3}+\frac{1}{6}+\frac{1}{9}=\frac{6}{18}+\frac{3}{18}+\frac{2}{18}=\frac{11}{18}$ and $3+6+9=18$ , so the answer is $\frac{11}{18}\cdot18=\boxed{11}$ .

Solution 3

$3 \left(\frac{1}{3}\right) + 3 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{9}\right)=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}$ $6 \left(\frac{1}{3}\right) + 6 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{9}\right)=2+1+\frac{2}{3}=\frac{22}{6}$ $9 \left(\frac{1}{3}\right) + 9 \left(\frac{1}{6}\right) + 9 \left(\frac{1}{9}\right)=3+\frac{3}{2}+1=\frac{33}{6}$ Therefore, the answer is $\frac{11}{6}+\frac{22}{6}+\frac{33}{6}=11$

- kante314 -

Solution 4

Solution by pog

Recall that, for any nonzero $a$ , $b$ , $c$ , we have that $\dfrac1a+\dfrac1b+\dfrac1c = \dfrac{\sum_{\text{sym}}ab}{abc}$ . By Vieta's, we see that $a$ , $b$ , and $c$ are the roots of the monic cubic polynomial $Q(x) = x^3 - px^2 + \left(\sum_{\text{sym}}ab\right) x - abc,$ where $p$ is an arbitrary constant equal to $a + b + c$ .

In the given equation, we have that $a = 3$ , $b = 6$ , and $c = 9$ , so $p = 3 + 6 + 9 = 18$ and $abc = 3 \cdot 6 \cdot 9 = 162$ , for $Q(x) = x^3 - 18x^2 + \left(\sum_{\text{sym}}ab\right) x - 162.$ (Finding the former of the latter two coefficients would lead to bashy casework, which we avoid in this extremely elegant solution.)

Since our cubic has roots $3$ , $6$ , and $9$ , the discriminant of our monic cubic polynomial $A^2B^2 + 18ABC - 4B^3 - 4A^3C - 27C^2$ (where $A$ , $B$ , and $C$ are the latter three coefficients of $Q(x)$ , respectively--do not confuse them for the roots of our polynomial equal to $a=3$ , $b=6$ , and $c=9$ , respectively) must not be equal to $0$ , or our cubic will have a double root, which it clearly does not (proof here is left to the reader).

Thus, substitution of the coefficients into our discriminant gives that $324 \cdot \left(\sum_{\text{sym}}ab\right)^2 + 324 \left(\sum_{\text{sym}}ab\right) \cdot 162 - 4\left(\sum_{\text{sym}}ab\right)^3 - 4 \cdot 18^3 \cdot 162 - 27 \cdot 162^2$ must not be equal to $0$ . Letting $\sum_{\text{sym}}ab = n$ , expanding gives that $324n^2 + 52488n - 4n^3 - 3779136 - 708588 = -4n^3 + 324n^2 + 52488n - 4487724$ is not equal to $0$ . Thus $n$ must not be a root of our cubic, so from here we apply the cubic formula to find that $n$ is not equal to (approximately) $-115.853202973303$ , $96.4974047104675$ , or $100.355798262835$ .

Recall that, from earlier, our answer is equal to $\left(\dfrac13+\dfrac16+\dfrac19\right)(3 + 6 + 9) = \left(\dfrac1a+\dfrac1b+\dfrac1c\right)(a+b+c) = \left(\dfrac{\sum_{\text{sym}}ab}{162}\right)(18) = \dfrac{n}{162} \cdot 18 = \dfrac{n}{9}$ . By bounding, $n$ is between the aforementioned values of $96.4974047104675$ and $100.355798262835$ , so since it must be a multiple of $9$ (all answers in the Junior Mathematicians' Problem Solving Competition are integers), it is equal to $99$ and the requested answer is equal to $\dfrac{99}{9} = \boxed{11}$ . This corroborates with jasperE3's strict bounding. $\blacksquare$

Figure 1: Final diagram from AoPS user awang11:

@@ Line 11: / Line 11: @@
 <math>\frac{1}{3}+\frac{1}{6}+\frac{1}{9}=\frac{6}{18}+\frac{3}{18}+\frac{2}{18}=\frac{11}{18}</math> and <math>3+6+9=18</math>, so the answer is <math>\frac{11}{18}\cdot18=\boxed{11}</math>.
+== Solution 3 ==
+<cmath>3 \left(\frac{1}{3}\right) + 3 \left(\frac{1}{6}\right) + 3 \left(\frac{1}{9}\right)=1+\frac{1}{2}+\frac{1}{3}=\frac{11}{6}</cmath><cmath>6 \left(\frac{1}{3}\right) + 6 \left(\frac{1}{6}\right) + 6 \left(\frac{1}{9}\right)=2+1+\frac{2}{3}=\frac{22}{6}</cmath><cmath>9 \left(\frac{1}{3}\right) + 9 \left(\frac{1}{6}\right) + 9 \left(\frac{1}{9}\right)=3+\frac{3}{2}+1=\frac{33}{6}</cmath>
+Therefore, the answer is <math>\frac{11}{6}+\frac{22}{6}+\frac{33}{6}=11</math>
+- kante314 -
+== Solution 4 ==
+''Solution by pog''
+Recall that, for any nonzero <math>a</math>, <math>b</math>, <math>c</math>, we have that <math>\dfrac1a+\dfrac1b+\dfrac1c = \dfrac{\sum_{\text{sym}}ab}{abc}</math>. By Vieta's, we see that <math>a</math>, <math>b</math>, and <math>c</math> are the roots of the monic cubic polynomial <cmath>Q(x) = x^3 - px^2 + \left(\sum_{\text{sym}}ab\right) x - abc,</cmath> where <math>p</math> is an arbitrary constant equal to <math>a + b + c</math>.
+In the given equation, we have that <math>a = 3</math>, <math>b = 6</math>, and <math>c = 9</math>, so <math>p = 3 + 6 + 9 = 18</math> and <math>abc = 3 \cdot 6 \cdot 9 = 162</math>, for <cmath>Q(x) =  x^3 - 18x^2 + \left(\sum_{\text{sym}}ab\right) x - 162.</cmath> (Finding the former of the latter two coefficients would lead to bashy casework, which we avoid in this extremely elegant solution.)
+Since our cubic has roots <math>3</math>, <math>6</math>, and <math>9</math>, the discriminant of our monic cubic polynomial <math>A^2B^2 + 18ABC - 4B^3 - 4A^3C - 27C^2</math> (where <math>A</math>, <math>B</math>, and <math>C</math> are the latter three coefficients of <math>Q(x)</math>, respectively--do not confuse them for the roots of our polynomial equal to <math>a=3</math>, <math>b=6</math>, and <math>c=9</math>, respectively) must not be equal to <math>0</math>, or our cubic will have a double root, which it clearly does not (proof here is left to the reader).
+Thus, substitution of the coefficients into our discriminant gives that <math>324 \cdot \left(\sum_{\text{sym}}ab\right)^2 + 324 \left(\sum_{\text{sym}}ab\right) \cdot 162 - 4\left(\sum_{\text{sym}}ab\right)^3 - 4 \cdot 18^3 \cdot 162 - 27 \cdot 162^2</math> must not be equal to <math>0</math>. Letting <math>\sum_{\text{sym}}ab = n</math>, expanding gives that <math>324n^2 + 52488n - 4n^3 - 3779136 - 708588 = -4n^3 + 324n^2 + 52488n - 4487724</math> is not equal to <math>0</math>. Thus <math>n</math> must not be a root of our cubic, so from here we apply the cubic formula to find that <math>n</math> is not equal to (approximately) <math>-115.853202973303</math>, <math>96.4974047104675</math>, or <math>100.355798262835</math>.
+Recall that, from earlier, our answer is equal to <math>\left(\dfrac13+\dfrac16+\dfrac19\right)(3 + 6 + 9) = \left(\dfrac1a+\dfrac1b+\dfrac1c\right)(a+b+c) = \left(\dfrac{\sum_{\text{sym}}ab}{162}\right)(18) = \dfrac{n}{162} \cdot 18 = \dfrac{n}{9}</math>. By bounding, <math>n</math> is between the aforementioned values of <math>96.4974047104675</math> and <math>100.355798262835</math>, so since it must be a multiple of <math>9</math> (all answers in the Junior Mathematicians' Problem Solving Competition are integers), it is equal to <math>99</math> and the requested answer is equal to <math>\dfrac{99}{9} = \boxed{11}</math>. This corroborates with jasperE3's strict bounding. <math>\blacksquare</math>
+<div style='text-align: center;'>'''Figure 1:''' Final diagram from AoPS user ''awang11'':
+[[File:Screen Shot 2021-07-12 at 10.25.44 AM.png||400px]]</div>
 ==See also==
@@ Line 16: / Line 40: @@
 #[[2021 JMPSC Sprint Answer Key|2021 JMPSC Sprint Answer Key]]
 #[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
 {{JMPSC Notice}}

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