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Difference between revisions of "2014 AIME I Problems/Problem 9"

(→‎Solution 1)
(→‎Solution 3)
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==Solution 3==
 
==Solution 3==
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Observing the equation, we notice that the coefficient for the middle term <math>-4029</math> is equal to <math>-2{\sqrt{2014}}^2-1</math>. Also notice that the coefficient for the <math>{x^3}</math> term is <math>\sqrt{2014}</math>. Therefore, if the original expression was to be factored into a linear binomial and a quadratic trinomial, the <math>x</math> term of the binomial would have a coefficient of <math>\sqrt{2014}</math>. Similarly, the <math>x</math> term of the trinomial would also have a coefficient of <math>\sqrt{2014}</math>. The factored form of the expression would look something like the following:
+
Observing the equation, we notice that the coefficient for the middle term <math>-4029</math> is equal to <cmath>-2{\sqrt{2014}}^2-1</cmath>.  
−
<math>({\sqrt{2014}}x-a)(x^2-n{\sqrt{2014}}x-b)</math> where <math>{a, b,c}</math> are all positive integers (because the <math>{x^2}</math> term of the original expression is negative, and the constant term is positive), and <math>{ab=2}</math>.
 
  
−
Multiplying this expression out gives <math>{{\sqrt{2014}x^3-(2014n+a)x^2+(an{\sqrt{2014}}-b{\sqrt{2014}})x+ab}}</math>. Equating this with the original expression gives <math>{2014n+a}=-4029</math>. The only positive integer solutions of this expression is <math>(n, a)=(1, 2015)</math> or <math>(2, 1)</math>. If <math>(n, a)=(1, 2015)</math> then setting <math>{an{\sqrt{2014}}-b{\sqrt{2014}}}=0</math> yields <math>{b=2015}</math> and therefore <math>{ab=2015^2}</math> which clearly isn't equal to <math>2</math> as the constant term. Therefore, <math>(n, a)=(2, 1)</math> and the factored form of the expression is:
+
Also notice that the coefficient for the <math>{x^3}</math> term is <math>\sqrt{2014}</math>. Therefore, if the original expression was to be factored into a linear binomial and a quadratic trinomial, the <math>x</math> term of the binomial would have a coefficient of <math>\sqrt{2014}</math>. Similarly, the <math>x</math> term of the trinomial would also have a coefficient of <math>\sqrt{2014}</math>. The factored form of the expression would look something like the following:
−
<math>({\sqrt{2014}}x-1)(x^2-2{\sqrt{2014}}x-2)</math>. Therefore, one of the three roots of the original expression is <math>{x=\dfrac{1}{\sqrt{2014}}}</math>.
+
<cmath>({\sqrt{2014}}x-a)(x^2-n{\sqrt{2014}}x-b)</cmath> where <math>{a, b,c}</math> are all positive integers (because the <math>{x^2}</math> term of the original expression is negative, and the constant term is positive), and <cmath>{ab=2}</cmath>
−
Using the quadratic formula yields the other two roots as <math>{x={\sqrt{2014}}+{\sqrt{2016}}}</math> and <math>{x={\sqrt{2014}}-{\sqrt{2016}}}</math>. Arranging the roots in ascending order (in the order <math>x_1<x_2<x_3</math>), <math>{\sqrt{2014}}-{\sqrt{2016}}<\dfrac{1}{\sqrt{2014}}<{\sqrt{2014}}+{\sqrt{2016}}</math>.
+
 
−
Therefore, <math>x_2(x_1+x_3)=\dfrac{1}{\sqrt{2014}}{2\sqrt{2014}}=\boxed{002}</math>.
+
Multiplying this expression out gives <cmath>{{\sqrt{2014}x^3-(2014n+a)x^2+(an{\sqrt{2014}}-b{\sqrt{2014}})x+ab}}</cmath> Equating this with the original expression gives <cmath>{2014n+a}=-4029</cmath> The only positive integer solutions of this expression is <math>(n, a)=(1, 2015)</math> or <math>(2, 1)</math>. If <math>(n, a)=(1, 2015)</math> then setting <math>{an{\sqrt{2014}}-b{\sqrt{2014}}}=0</math> yields <math>{b=2015}</math> and therefore <math>{ab=2015^2}</math> which clearly isn't equal to <math>2</math> as the constant term. Therefore, <math>(n, a)=(2, 1)</math> and the factored form of the expression is:
 +
<cmath>({\sqrt{2014}}x-1)(x^2-2{\sqrt{2014}}x-2)</cmath> Therefore, one of the three roots of the original expression is <cmath>{x=\dfrac{1}{\sqrt{2014}}}</cmath>
 +
Using the quadratic formula yields the other two roots as <cmath>{x={\sqrt{2014}}+{\sqrt{2016}}}</cmath> and <cmath>{x={\sqrt{2014}}-{\sqrt{2016}}}</cmath> Arranging the roots in ascending order (in the order <math>x_1<x_2<x_3</math>), <cmath>{\sqrt{2014}}-{\sqrt{2016}}<\dfrac{1}{\sqrt{2014}}<{\sqrt{2014}}+{\sqrt{2016}}</cmath>
 +
Therefore, <cmath>x_2(x_1+x_3)=\dfrac{1}{\sqrt{2014}}{2\sqrt{2014}}=\boxed{002}</cmath>
  
 
==Solution 4==
 
==Solution 4==

Revision as of 11:33, 29 August 2021

Problem 9

Let $x_1<x_2<x_3$ be the three real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $x_2(x_1+x_3)$.

Solution 1

Substituting $n$ for $2014$, we get \[\sqrt{n}x^3 - (1+2n)x^2 + 2 = \sqrt{n}x^3 - x^2 - 2nx^2 + 2\] \[= x^2(\sqrt{n}x - 1) - 2(nx^2 - 1) = 0\] Noting that $nx^2 - 1$ factors as a difference of squares to \[(\sqrt{n}x - 1)(\sqrt{n}x+1)\] we can factor the left side as \[(\sqrt{n}x - 1)(x^2 - 2(\sqrt{n}x+1))\] This means that $\frac{1}{\sqrt{n}}$ is a root, and the other two roots are the roots of $x^2 - 2\sqrt{n}x - 2$. Note that the constant term of the quadratic is negative, so one of the two roots is positive and the other is negative. In addition, by Vieta's Formulas, the roots sum to $2\sqrt{n}$, so the positive root must be greater than $2\sqrt{n}$ in order to produce this sum when added to a negative value. Since $0 < \frac{1}{\sqrt{2014}} < 2\sqrt{2014}$ is clearly true, $x_2 = \frac{1}{\sqrt{2014}}$ and $x_1 + x_3 = 2\sqrt{2014}$. Multiplying these values together, we find that $x_2(x_1+x_3) = \boxed{002}$.

Solution 2

From Vieta's formulae, we know that \[x_1x_2x_3 = \dfrac{-2}{\sqrt{2014}}\] \[x_1 + x_2 + x_3 = \dfrac{4029}{\sqrt{2014}}\] and \[x_1x_2 + x_2x_3 + x_1x_3 = 0\] Thus, we know that \[x_2(x_1 + x_3) = -x_1x_3\]

Now consider the polynomial with roots $x_1x_2, x_2x_3,$ and $x_1x_3$. Expanding the polynomial \[(x - x_1x_2)(x - x_2x_3)(x - x_1x_3)\]we get the polynomial \[x^3 - (x_1x_2 + x_2x_3 + x_1x_3)x^2 + (x_1x_2x_3)(x_1 + x_2 + x_3)x - (x_1x_2x_3)^2\] Substituting the values obtained from Vieta's formulae, we find that this polynomial is \[x^3 - \dfrac{8058}{2014}x - \dfrac{4}{2014}\] We know $x_1x_3$ is a root of this polynomial, so we set it equal to 0 and simplify the resulting expression to \[1007x^3 - 4029x - 2 = 0\]

Given the problem conditions, we know there must be at least 1 integer solution, and that it can't be very large (because the $x^3$ term quickly gets much larger/smaller than the other 2). Trying out some numbers, we quickly find that $x = -2$ is a solution. Factoring it out, we get that \[1007x^3 - 4029x - 2 = (x+2)(1007x^2 - 2014x - 1)\] Since the other quadratic factor clearly does not have any integer solutions and since the AIME has only positive integer answers, we know that this must be the answer they are looking for. Thus, \[x_1x_3 = -2\] so \[-x_1x_3 = \boxed{002}\]and we're done.

Solution 3

Observing the equation, we notice that the coefficient for the middle term $-4029$ is equal to \[-2{\sqrt{2014}}^2-1\].

Also notice that the coefficient for the ${x^3}$ term is $\sqrt{2014}$. Therefore, if the original expression was to be factored into a linear binomial and a quadratic trinomial, the $x$ term of the binomial would have a coefficient of $\sqrt{2014}$. Similarly, the $x$ term of the trinomial would also have a coefficient of $\sqrt{2014}$. The factored form of the expression would look something like the following: \[({\sqrt{2014}}x-a)(x^2-n{\sqrt{2014}}x-b)\] where ${a, b,c}$ are all positive integers (because the ${x^2}$ term of the original expression is negative, and the constant term is positive), and \[{ab=2}\]

Multiplying this expression out gives \[{{\sqrt{2014}x^3-(2014n+a)x^2+(an{\sqrt{2014}}-b{\sqrt{2014}})x+ab}}\] Equating this with the original expression gives \[{2014n+a}=-4029\] The only positive integer solutions of this expression is $(n, a)=(1, 2015)$ or $(2, 1)$. If $(n, a)=(1, 2015)$ then setting ${an{\sqrt{2014}}-b{\sqrt{2014}}}=0$ yields ${b=2015}$ and therefore ${ab=2015^2}$ which clearly isn't equal to $2$ as the constant term. Therefore, $(n, a)=(2, 1)$ and the factored form of the expression is: \[({\sqrt{2014}}x-1)(x^2-2{\sqrt{2014}}x-2)\] Therefore, one of the three roots of the original expression is \[{x=\dfrac{1}{\sqrt{2014}}}\] Using the quadratic formula yields the other two roots as \[{x={\sqrt{2014}}+{\sqrt{2016}}}\] and \[{x={\sqrt{2014}}-{\sqrt{2016}}}\] Arranging the roots in ascending order (in the order $x_1<x_2<x_3$), \[{\sqrt{2014}}-{\sqrt{2016}}<\dfrac{1}{\sqrt{2014}}<{\sqrt{2014}}+{\sqrt{2016}}\] Therefore, \[x_2(x_1+x_3)=\dfrac{1}{\sqrt{2014}}{2\sqrt{2014}}=\boxed{002}\]

Solution 4

By Vieta's, we are seeking to find $x_2(x_1+x_3)=x_1x_2+x_2x_3=-x_1x_3=\frac{2}{\sqrt{2014}x_2}$. Substitute $n=-x_1x_3$ and $x_2=\frac{2}{\sqrt{2014}n}$. Substituting this back into the original equation, we have $\frac{4}{1007n^3}-\frac{8058}{1007n^2}+2=0$, so $2n^3-\frac{8058}{1007}n+\frac{4}{1007}=2n^3-\frac{8058n-4}{1007}=0$. Hence, $8058n-4\equiv 2n-4 \equiv 0 \pmod{1007}$, and $n\equiv 2\pmod{1007}$. But since $n\le 999$ because it is our desired answer, the only possible value for $n$ is $\boxed{002}$ BEST PROOOFFFF Stormersyle & mathleticguyyy


Solution 5

Let $y =\frac{x}{\sqrt{2014}}.$ The original equation simplifies to $\frac{y^3}{2014} -\frac{4029y^2}{2014}+2 = 0 \implies y^3 - 4029y + 4028=0.$ Here we clearly see that $y=1$ is a root. Divideing $y-1$ from the sum we find that $(y-1)(y^2-4028y-4028)=0.$ From simple bounding we see that $y=1$ is the middle root. Therefore $x_{2}(x_{1}+x_{3}) =\frac{1}{\sqrt{2014}} \cdot\frac{4028}{\sqrt{2014}} = \boxed{002}.$

Solution 6

$\sqrt{2014}$ occurs multiple times, so let k = $\sqrt{2014}$.

The equation becomes $0 = kx^3 - (2k^2 + 1)x^2 + 2$. Since we want to relate k and x, we should solve for one of them. We can't solve for x, since that would require the cubic formula, so we solve for k, and express it in terms of a quadratic, and apply the quadratic formula.

We get the roots are:

$y = \frac{1}{x}$, and $y = \frac{x}{2} - \frac{1}{x}$.

In the first case, $x = \frac{1}{y} = \frac{1}{\sqrt{2014}}$.

In the second case, $x^2 - 2\sqrt{2014} - 2 = 0$. The solutions are $\sqrt{2014} \pm \sqrt{2016}$. The sum of these 2 solutions is $2 \sqrt{2014}$, and $\frac{1}{\sqrt{2014}}$ is the middle solution, and thus, $(x_1 + x_3) \cdot x_2 = 2$


See also

2014 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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