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Difference between revisions of "2020 AIME II Problems/Problem 11"

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==Solution 3==
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==Problem==
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Let <math>P(x) = x^2 - 3x - 7</math>, and let <math>Q(x)</math> and <math>R(x)</math> be two quadratic polynomials also with the coefficient of <math>x^2</math> equal to <math>1</math>. David computes each of the three sums <math>P + Q</math>, <math>P + R</math>, and <math>Q + R</math> and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If <math>Q(0) = 2</math>, then <math>R(0) = \frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>.
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==Solution 1==
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Let <math>Q(x) = x^2 + ax + 2</math> and <math>R(x) = x^2 + bx + c</math>.  We can write the following:
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<cmath>P + Q = 2x^2 + (a - 3)x - 5</cmath>
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<cmath>P + R = 2x^2 + (b - 3)x + (c - 7)</cmath>
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<cmath>Q + R = 2x^2 + (a + b)x + (c + 2)</cmath>
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Let the common root of <math>P+Q,P+R</math> be <math>r</math>; <math>P+R,Q+R</math> be <math>s</math>; and <math>P+Q,Q+R</math> be <math>t</math>.  We then have that the roots of <math>P+Q</math> are <math>r,t</math>, the roots of <math>P + R</math> are <math>r, s</math>, and the roots of <math>Q + R</math> are <math>s,t</math>.
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By Vieta's, we have:
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<cmath> r + t = \dfrac{3 - a}{2}\tag{1}</cmath>
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<cmath>r + s = \dfrac{3 - b}{2}\tag{2}</cmath>
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<cmath>s + t = \dfrac{-a - b}{2}\tag{3}</cmath>
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<cmath>rt = \dfrac{-5}{2}\tag{4}</cmath>
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<cmath>rs = \dfrac{c - 7}{2}\tag{5}</cmath>
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<cmath>st = \dfrac{c + 2}{2}\tag{6}</cmath>
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Subtracting <math>(3)</math> from <math>(1)</math>, we get <math>r - s = \dfrac{3 + b}{2}</math>.  Adding this to <math>(2)</math>, we get <math>2r = 3 \implies r = \dfrac{3}{2}</math>.  This gives us that <math>t = \dfrac{-5}{3}</math> from <math>(4)</math>.  Substituting these values into <math>(5)</math> and <math>(6)</math>, we get <math>s = \dfrac{c-7}{3}</math> and <math>s = \dfrac{-3c - 6}{10}</math>.  Equating these values, we get <math>\dfrac{c-7}{3} = \dfrac{-3c-6}{10} \implies c = \dfrac{52}{19} = R(0)</math>.  Thus, our answer is <math>52 + 19 = \boxed{071}</math>. ~ TopNotchMath
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==Solution 2==
  
 
We know that <math>P(x)=x^2-3x-7</math>.  
 
We know that <math>P(x)=x^2-3x-7</math>.  
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We could work out the system of equations, but it's pretty easy to see that <math>p=\frac32, q=-\frac{a}{2}, r=-\frac{b}{2}</math>.
 
We could work out the system of equations, but it's pretty easy to see that <math>p=\frac32, q=-\frac{a}{2}, r=-\frac{b}{2}</math>.
  
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<math>\text{Again, by vietas, }pq=-\frac52\text{, } pr=\frac{c-7}{2}\text{, } qr=\frac{c+2}{2}</math>
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<math>\text{Again, by vietas, }pq=-\frac52\text{, } pr=\frac{c-7}{2}\text{, } qr=\frac{c+2}{2}\text{, } \text{multiplying everything together a}\text{nd taking the sqrt of both sides,}</math>
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<math>\text{Multiplying everything together a}\text{nd then taking the sqrt of both sides,}</math>
 
 
<cmath>(pqr)^2=\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)</cmath>
 
<cmath>(pqr)^2=\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)</cmath>
 
<cmath>pqr=\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)} </cmath>
 
<cmath>pqr=\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)} </cmath>
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<math>\text{Now, we divide this }\text{equation by }qr=\frac{c+2}{2} </math>
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<math>\text{Dividing this }\text{equation by }qr=\frac{c+2}{2} </math>
 
<cmath>\frac{pqr}{qr}=\frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)}}{\frac{c+2}{2}} </cmath>
 
<cmath>\frac{pqr}{qr}=\frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)}}{\frac{c+2}{2}} </cmath>
 
<cmath>p = \frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}}{\sqrt{\frac{c+2}{2}}} </cmath>
 
<cmath>p = \frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}}{\sqrt{\frac{c+2}{2}}} </cmath>
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<cmath>\frac94=\frac{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}{\frac{c+2}{2}} </cmath>
 
<cmath>\frac94=\frac{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}{\frac{c+2}{2}} </cmath>
 
<math>\text{Solving gives } c=\frac{52}{19}, \text{ so our answer is }\boxed{071}</math>
 
<math>\text{Solving gives } c=\frac{52}{19}, \text{ so our answer is }\boxed{071}</math>
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\end{align*}
 
  
 
~quacker88
 
~quacker88
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==Solution 3 (Official MAA)==
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Let the common root of <math>P+Q</math> and <math>P+R</math> be <math>p</math>, the common root of <math>P+Q</math> and <math>Q+R</math> be <math>q</math>, and the common root of <math>Q+R</math> and <math>P+R</math> be <math>r</math>. Because <math>p</math> and <math>q</math> are both roots of <math>P+Q</math> and <math>P+Q</math> has leading coefficient <math>2</math>, it follows that <math>P(x) + Q(x) = 2(x-p)(x-q).</math> Similarly, <math>P(x) + R(x) = 2(x-p)(x-r)</math> and <math>Q(x) + R(x) = 2(x-q)(x-r)</math>. Adding these three equations together and dividing by <math>2</math> yields<cmath>P(x) + Q(x) + R(x) = (x-p)(x-q) + (x-p)(x-r) + (x-q)(x-r),</cmath>so
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<cmath>P(x) = (P(x) + Q(x) + R(x)) - (Q(x) + R(x))</cmath>
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<cmath>= (x-p)(x-q) + (x-p)(x-r) - (x-q)(x-r) </cmath>
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<cmath>= x^2 - 2px + (pq + pr - qr).</cmath>
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Similarly,
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<cmath>Q(x) =  x^2 - 2qx + (pq + qr - pr) \text{~ and}</cmath>
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<cmath>R(x) =  x^2 - 2rx + (pr + qr - pq).</cmath>
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Comparing the <math>x</math> coefficients yields <math>p = \tfrac32</math>, and comparing the constant coefficients yields <math>-7 = pq + pr - qr = \tfrac32(q+r) - qr</math>. The fact that <math>Q(0) = 2</math> implies that <math>\tfrac32(q-r) + qr = 2</math>. Adding these two equations yields <math>q = -\tfrac53</math>, and so substituting back in to solve for <math>r</math> gives <math>r=-\tfrac{27}{19}</math>. Finally,<cmath>R(0) = pr + qr - pq = \left(-\frac{27}{19}\right)\left(\frac32-\frac53\right) + \frac52 = \frac{9}{38} + \frac52 = \frac{52}{19}.</cmath>The requested sum is <math>52 + 19 = 71</math>. Note that <math>Q(x) = x^2 + \frac{10}3x + 2</math> and <math>R(x) = x^2 + \frac{54}{19}x + \frac{52}{19}</math>.
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==Video Solution==
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https://youtu.be/BQlab3vjjxw ~ CNCM
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 +
Another one:
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https://www.youtube.com/watch?v=AXN9x51KzNI
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 +
==See Also==
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{{AIME box|year=2020|n=II|num-b=10|num-a=12}}
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{{MAA Notice}}
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 +
[[Category:Intermediate Algebra Problems]]

Latest revision as of 20:16, 28 December 2023

Problem

Let $P(x) = x^2 - 3x - 7$, and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$. David computes each of the three sums $P + Q$, $P + R$, and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are distinct. If $Q(0) = 2$, then $R(0) = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Solution 1

Let $Q(x) = x^2 + ax + 2$ and $R(x) = x^2 + bx + c$. We can write the following: \[P + Q = 2x^2 + (a - 3)x - 5\] \[P + R = 2x^2 + (b - 3)x + (c - 7)\] \[Q + R = 2x^2 + (a + b)x + (c + 2)\] Let the common root of $P+Q,P+R$ be $r$; $P+R,Q+R$ be $s$; and $P+Q,Q+R$ be $t$. We then have that the roots of $P+Q$ are $r,t$, the roots of $P + R$ are $r, s$, and the roots of $Q + R$ are $s,t$.

By Vieta's, we have: \[r + t = \dfrac{3 - a}{2}\tag{1}\] \[r + s = \dfrac{3 - b}{2}\tag{2}\] \[s + t = \dfrac{-a - b}{2}\tag{3}\] \[rt = \dfrac{-5}{2}\tag{4}\] \[rs = \dfrac{c - 7}{2}\tag{5}\] \[st = \dfrac{c + 2}{2}\tag{6}\]

Subtracting $(3)$ from $(1)$, we get $r - s = \dfrac{3 + b}{2}$. Adding this to $(2)$, we get $2r = 3 \implies r = \dfrac{3}{2}$. This gives us that $t = \dfrac{-5}{3}$ from $(4)$. Substituting these values into $(5)$ and $(6)$, we get $s = \dfrac{c-7}{3}$ and $s = \dfrac{-3c - 6}{10}$. Equating these values, we get $\dfrac{c-7}{3} = \dfrac{-3c-6}{10} \implies c = \dfrac{52}{19} = R(0)$. Thus, our answer is $52 + 19 = \boxed{071}$. ~ TopNotchMath

Solution 2

We know that $P(x)=x^2-3x-7$.

Since $Q(0)=2$, the constant term in $Q(x)$ is $2$. Let $Q(x)=x^2+ax+2$.

Finally, let $R(x)=x^2+bx+c$.

$P(x)+Q(x)=2x^2+(a-3)x-5$. Let its roots be $p$ and $q$.

$P(x)+R(x)=2x^2+(b-3)x+(c-7)$ Let its roots be $p$ and $r$.

$Q(x)+R(x)=2x^2+(a+b)x+(c+2)$. Let its roots be $q$ and $r$.

By vietas, $p+q=\frac{3-a}{2}, p+r=\frac{3-b}{2}, q+r=\frac{-(a+b)}{2}$

We could work out the system of equations, but it's pretty easy to see that $p=\frac32, q=-\frac{a}{2}, r=-\frac{b}{2}$.

$\text{Again, by vietas, }pq=-\frac52\text{, } pr=\frac{c-7}{2}\text{, } qr=\frac{c+2}{2}\text{, } \text{multiplying everything together a}\text{nd taking the sqrt of both sides,}$ \[(pqr)^2=\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)\] \[pqr=\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)}\] $\text{Dividing this }\text{equation by }qr=\frac{c+2}{2}$ \[\frac{pqr}{qr}=\frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)\left(\frac{c+2}{2}\right)}}{\frac{c+2}{2}}\] \[p = \frac{\sqrt{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}}{\sqrt{\frac{c+2}{2}}}\] $\text{Recall th}\text{at }p=\frac32 \text{ and square both sides}$ \[\frac94=\frac{\left(-\frac52\right)\left(\frac{c-7}{2}\right)}{\frac{c+2}{2}}\] $\text{Solving gives } c=\frac{52}{19}, \text{ so our answer is }\boxed{071}$

~quacker88

Solution 3 (Official MAA)

Let the common root of $P+Q$ and $P+R$ be $p$, the common root of $P+Q$ and $Q+R$ be $q$, and the common root of $Q+R$ and $P+R$ be $r$. Because $p$ and $q$ are both roots of $P+Q$ and $P+Q$ has leading coefficient $2$, it follows that $P(x) + Q(x) = 2(x-p)(x-q).$ Similarly, $P(x) + R(x) = 2(x-p)(x-r)$ and $Q(x) + R(x) = 2(x-q)(x-r)$. Adding these three equations together and dividing by $2$ yields\[P(x) + Q(x) + R(x) = (x-p)(x-q) + (x-p)(x-r) + (x-q)(x-r),\]so \[P(x) = (P(x) + Q(x) + R(x)) - (Q(x) + R(x))\] \[= (x-p)(x-q) + (x-p)(x-r) - (x-q)(x-r)\] \[= x^2 - 2px + (pq + pr - qr).\] Similarly, \[Q(x) = x^2 - 2qx + (pq + qr - pr) \text{~ and}\] \[R(x) = x^2 - 2rx + (pr + qr - pq).\] Comparing the $x$ coefficients yields $p = \tfrac32$, and comparing the constant coefficients yields $-7 = pq + pr - qr = \tfrac32(q+r) - qr$. The fact that $Q(0) = 2$ implies that $\tfrac32(q-r) + qr = 2$. Adding these two equations yields $q = -\tfrac53$, and so substituting back in to solve for $r$ gives $r=-\tfrac{27}{19}$. Finally,\[R(0) = pr + qr - pq = \left(-\frac{27}{19}\right)\left(\frac32-\frac53\right) + \frac52 = \frac{9}{38} + \frac52 = \frac{52}{19}.\]The requested sum is $52 + 19 = 71$. Note that $Q(x) = x^2 + \frac{10}3x + 2$ and $R(x) = x^2 + \frac{54}{19}x + \frac{52}{19}$.

Video Solution

https://youtu.be/BQlab3vjjxw ~ CNCM

Another one:

https://www.youtube.com/watch?v=AXN9x51KzNI

See Also

2020 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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All AIME Problems and Solutions

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