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Difference between revisions of "2014 AIME II Problems/Problem 5"

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==Problem 5==
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==Problem==
 
<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Real numbers <math>r</math> and <math>s</math> are roots of <math>p(x)=x^3+ax+b</math>, and <math>r+4</math> and <math>s-3</math> are roots of <math>q(x)=x^3+ax+b+240</math>. Find the sum of all possible values of <math>|b|</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
 
<!-- don't remove the following tag, for PoTW on the Wiki front page--><onlyinclude>Real numbers <math>r</math> and <math>s</math> are roots of <math>p(x)=x^3+ax+b</math>, and <math>r+4</math> and <math>s-3</math> are roots of <math>q(x)=x^3+ax+b+240</math>. Find the sum of all possible values of <math>|b|</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude>
  
 
==Solution 1==
 
==Solution 1==
 +
Because the coefficient of <math>x^2</math> in both <math>p(x)</math> and <math>q(x)</math> is 0, the remaining root of <math>p(x)</math> is <math>-(r+s)</math>, and the remaining root of <math>q(x)</math> is <math>-(r+s+1)</math>. The coefficients of <math>x</math> in <math>p(x)</math> and <math>q(x)</math> are both equal to <math>a</math>, and equating the two coefficients gives
 +
<cmath>rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2 </cmath>from which <math>s = \tfrac 12 (5r+13)</math>. The product of the roots of <math>p(x)</math> differs from that of <math>q(x)</math> by <math>240</math>, so<cmath>(r+4)\cdot \tfrac 12 (5r+7)\cdot \tfrac 12(7r+15)- r\cdot \tfrac 12 (5r+13)\cdot \tfrac 12(7r+13)=240</cmath>from which <math>r^2+4r-5 =0</math>, with roots <math>r=1</math> and <math>r=-5</math>.
 +
 +
If <math>r = 1</math>, then the roots of <math>p(x)</math> are <math>r=1</math>, <math>s=9</math>, and <math>-(r+s)=-10</math>, and <math>b=-rst=90</math>.
 +
 +
If <math>r = -5</math>, then the roots of <math>p(x)</math> are <math>r=-5</math>, <math>s=-6</math>, and <math>-(r+s)=11</math>, and <math>b=-rst=-330</math>.
 +
 +
Therefore the requested sum is <math>|- 330| + |90| = \boxed{420}</math>.
 +
 +
==Solution 2==
 
Let <math>r</math>, <math>s</math>, and <math>-r-s</math> be the roots of <math>p(x)</math> (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for <math>s</math>. Also,
 
Let <math>r</math>, <math>s</math>, and <math>-r-s</math> be the roots of <math>p(x)</math> (per Vieta's). Then <math>r^3 + ar + b = 0</math> and similarly for <math>s</math>. Also,
 
<cmath>q(r+4) = (r+4)^3 + a(r+4) + b  + 240 = 12r^2 + 48r + 304 + 4a = 0</cmath>
 
<cmath>q(r+4) = (r+4)^3 + a(r+4) + b  + 240 = 12r^2 + 48r + 304 + 4a = 0</cmath>
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Simplifying and adding the equations gives
 
Simplifying and adding the equations gives
−
<cmath>3r^2 - 3s^2 + 12r + 9s + 147 = 0</cmath>
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<cmath>\begin{align}\tag{*}
 +
r^2 - s^2 + 4r + 3s + 49 &= 0
 +
\end{align}</cmath>
 +
Now, let's deal with the <math>ax</math> terms. Plugging the roots <math>r</math>, <math>s</math>, and <math>-r-s</math> into <math>p(x)</math> yields a long polynomial, and plugging the roots <math>r+4</math>, <math>s-3</math>, and <math>-1-r-s</math> into <math>q(x)</math> yields another long polynomial. Equating the coefficients of <math>x</math> in both polynomials, we get:
 +
<cmath>rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1),</cmath>
 +
which eventually simplifies to
 +
<cmath>s = \frac{13 + 5r}{2}.</cmath>
 +
Substitution into (*) should give <math>r = -5</math> and <math>r = 1</math>, corresponding to <math>s = -6</math> and <math>s = 9</math>, and <math>|b| = 330, 90</math>, for an answer of <math>\boxed{420}</math>.
 +
 
 +
==Solution 3==
 +
The roots of <math>p(x)</math> are <math>r</math>, <math>s</math>, and <math>-r-s</math> since they sum to <math>0</math> by Vieta's Formula (co-efficient of <math>x^2</math> term is <math>0</math>).
 +
 
 +
Similarly, the roots of <math>q(x)</math> are <math>r + 4</math>, <math>s - 3</math>, and <math>-r-s-1</math>, as they too sum to <math>0</math>.
 +
 
 +
Then:
 +
 
 +
<math>a = rs+r(-r-s)+s(-r-s) = rs-(r+s)^2</math> and <math>-b = rs(-r-s)</math> from <math>p(x)</math> and
 +
 
 +
<math>a=(r+4)(s-3)+(r+4)(-r-s-1)+(s-3)(-r-s-1) = (r+4)(s-3)-(r+s+1)^2</math> and <math>-(b+240)=(r+4)(s-3)(-r-s-1)</math> from <math>q(x)</math>.
 +
 
 +
From these equations, we can write that
 +
<cmath>rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2 = a</cmath>
 +
and simplifying gives
 +
<cmath>2s-5r-13=0 \Rightarrow s = \frac{5r+13}{2}.</cmath>
  
−
<cmath>r^2 - s^2 + 4r + 3s + 49 = 0 (*)</cmath>
 
  
−
Now, let's deal with the <math>ax</math> terms. Plugging the roots <math>r</math>, <math>s</math>, and <math>-r-s</math> into <math>p(x)</math> yields a long polynomial, and plugging the roots <math>r+4</math>, <math>s-3</math>, and <math>-1-r-s</math> into <math>q(x)</math> yields another long polynomial. Equating the coefficients of x in both polynomials:
 
−
<cmath>rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1),</cmath>
 
−
which eventually simplifies to
 
  
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<cmath>s = \frac{13 + 5r}{2}.</cmath>
+
We now move to the other two equations regarding the product of the roots. We see that we can cancel a negative from both sides to get
 +
<cmath>rs(r+s) = b</cmath>
 +
<cmath>(r+4)(s-3)(r+s+1)=b + 240.</cmath> Subtracting the first equation from the second equation gives us <math>(r+4)(s-3)(r+s+1) - rs(r+s) = 240</math>.
 +
 
 +
Expanding, simplifying, substituting <math>s = \frac{5r+13}{2}</math>, and simplifying some more yields the simple quadratic <math>r^2 + 4r - 5 = 0</math>, so <math>r = -5, 1</math>. Then <math>s = -6, 9</math>.
 +
 
 +
Finally, we substitute back into <math>b=rs(r+s)</math> to get <math>b = (-5)(-6)(-5-6) = -330</math>, or <math>b = (1)(9)(1 + 9) = 90</math>.
  
−
Substitution into (*) should give <math>r = -5</math> and <math>r = 1</math>, corresponding to <math>s = -6</math> and <math>s = 9</math>, and <math>|b| = 330, 90</math>, for an answer of <math>\boxed{420}</math>.
+
The answer is <math>|-330|+|90| = \boxed{420}</math>.
  
−
==Solution 2==
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==Solution 4==
−
As above, we know from Vieta's that the roots of <math>p(x)</math> are <math>r</math>, <math>s</math>, and <math>-r-s</math>. Similarly, the roots of <math>q(x)</math> are <math>r + 4</math>, <math>s - 3</math>, and <math>-r-s-1</math>. Then <math>rs+r(-r-s)+s(-r-s) = rs-(r+s)^2 = a</math> and <math>rs(-r-s) = -b</math> from <math>p(x)</math> and <math>(r+4)(s-3)+(r+4)(-r-s-1)+(s-3)(-r-s-1) = (r+4)(s-3)-(r+s+1)^2 = a</math> and <math>(r+4)(s-3)(-r-s-1)=-(b + 240)</math> from <math>q(x)</math>.
 
  
−
From these equations, we can write that <math>rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2</math>, and simplifying gives us <math>2s-5r-13=0</math> or <math>s = \frac{5r+13}{2}</math>.
+
By Vieta's, we know that the sum of roots of <math>p(x)</math> is <math>0</math>. Therefore,
 +
the roots of <math>p</math> are <math>r, s, -r-s</math>. By similar reasoning, the roots of <math>q(x)</math>
 +
are <math>r + 4, s - 3, -r - s - 1</math>. Thus, <math>p(x) = (x - r)(x - s)(x + r + s)</math>
 +
and <math>q(x) = (x - r - 4)(x - s + 3)(x + r + s + 1)</math>.
  
−
We now move to the other two equations. We see that we can cancel a negative from both sides to get <math>rs(r+s) = b</math> and <math>(r+4)(s-3)(r+s+1)=b + 240</math>. Subtracting the first from the second equation gives us <math>(r+4)(s-3)(r+s+1) - rs(r+s) = 240</math>. Expanding and simplifying, substituting <math>s = \frac{5r+13}{2}</math> and simplifying some more yields the simple quadratic <math>r^2 + 4r - 5 = 0</math>, so <math>r = -5, 1</math>. Then <math>s = -6, 9</math>.
+
Since <math>p(x)</math> and <math>q(x)</math> have the same coefficient for <math>x</math>, we can go ahead
 +
and match those up to get
 +
<cmath>\begin{align*}
 +
    rs - r(r + s) - s(r + s) &= (r + 4)(s - 3) - (r + 4)(r + s + 1) - (s - 3)(r + s + 1) \\
 +
    0 &= -13 - 5r + 2s \\
 +
    s &= \frac{5r + 13}{2}
 +
\end{align*}</cmath>
  
−
Finally, we substitute back in to get <math>b = (-5)(-6)(-5-6) = -330</math> or <math>b = (1)(9)(1 + 9) = 90</math>. Then the answer is <math>|-330|+|90| = \boxed{420}</math>.
+
At this point, we can go ahead and compare the constant term in <math>p(x)</math> and
 +
<math>q(x)</math>. Doing so is certainly valid, but we can actually do this another way. Notice that <math>p(s) = 0</math>. Therefore, <math>q(s) = 240</math>. If we plug that into
 +
our expression, we get that
 +
<cmath>\begin{align*}
 +
    q(s)  &= 3(s - r - 4)(r + 2s + 1) \\
 +
    240 &= 3(s - r - 4)(r + 2s + 1) \\
 +
    240 &= 3\left( \frac{3r + 5}{2} \right)(6r + 14) \\
 +
    80 &= (3r + 5)(3r + 7) \\
 +
    0 &= r^2 + 4r - 5
 +
\end{align*}</cmath>
 +
This tells us that <math>(r, s) = (1, 9)</math> or <math>(-5, -6)</math>. Since <math>-b</math> is the product of the roots, we have that the two possibilities are <math>1 \cdot 9 \cdot (-10) = -90</math>
 +
and <math>(-5)(-6)11 = 330</math>. Adding the absolute values of these gives us
 +
<math>\boxed{420}</math>.
  
 
== See also ==
 
== See also ==

Revision as of 14:51, 31 December 2023

Problem

Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.

Solution 1

Because the coefficient of $x^2$ in both $p(x)$ and $q(x)$ is 0, the remaining root of $p(x)$ is $-(r+s)$, and the remaining root of $q(x)$ is $-(r+s+1)$. The coefficients of $x$ in $p(x)$ and $q(x)$ are both equal to $a$, and equating the two coefficients gives \[rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2\]from which $s = \tfrac 12 (5r+13)$. The product of the roots of $p(x)$ differs from that of $q(x)$ by $240$, so\[(r+4)\cdot \tfrac 12 (5r+7)\cdot \tfrac 12(7r+15)- r\cdot \tfrac 12 (5r+13)\cdot \tfrac 12(7r+13)=240\]from which $r^2+4r-5 =0$, with roots $r=1$ and $r=-5$.

If $r = 1$, then the roots of $p(x)$ are $r=1$, $s=9$, and $-(r+s)=-10$, and $b=-rst=90$.

If $r = -5$, then the roots of $p(x)$ are $r=-5$, $s=-6$, and $-(r+s)=11$, and $b=-rst=-330$.

Therefore the requested sum is $|- 330| + |90| = \boxed{420}$.

Solution 2

Let $r$, $s$, and $-r-s$ be the roots of $p(x)$ (per Vieta's). Then $r^3 + ar + b = 0$ and similarly for $s$. Also, \[q(r+4) = (r+4)^3 + a(r+4) + b + 240 = 12r^2 + 48r + 304 + 4a = 0\]

Set up a similar equation for $s$:

\[q(s-3) = (s-3)^3 + a(s-3) + b + 240 = -9s^2 + 27s + 213 - 3a = 0.\]

Simplifying and adding the equations gives \begin{align}\tag{*} r^2 - s^2 + 4r + 3s + 49 &= 0 \end{align} Now, let's deal with the $ax$ terms. Plugging the roots $r$, $s$, and $-r-s$ into $p(x)$ yields a long polynomial, and plugging the roots $r+4$, $s-3$, and $-1-r-s$ into $q(x)$ yields another long polynomial. Equating the coefficients of $x$ in both polynomials, we get: \[rs + (-r-s)(r+s) = (r+4)(s-3) + (-r-s-1)(r+s+1),\] which eventually simplifies to \[s = \frac{13 + 5r}{2}.\] Substitution into (*) should give $r = -5$ and $r = 1$, corresponding to $s = -6$ and $s = 9$, and $|b| = 330, 90$, for an answer of $\boxed{420}$.

Solution 3

The roots of $p(x)$ are $r$, $s$, and $-r-s$ since they sum to $0$ by Vieta's Formula (co-efficient of $x^2$ term is $0$).

Similarly, the roots of $q(x)$ are $r + 4$, $s - 3$, and $-r-s-1$, as they too sum to $0$.

Then:

$a = rs+r(-r-s)+s(-r-s) = rs-(r+s)^2$ and $-b = rs(-r-s)$ from $p(x)$ and

$a=(r+4)(s-3)+(r+4)(-r-s-1)+(s-3)(-r-s-1) = (r+4)(s-3)-(r+s+1)^2$ and $-(b+240)=(r+4)(s-3)(-r-s-1)$ from $q(x)$.

From these equations, we can write that \[rs-(r+s)^2 = (r+4)(s-3)-(r+s+1)^2 = a\] and simplifying gives \[2s-5r-13=0 \Rightarrow s = \frac{5r+13}{2}.\]


We now move to the other two equations regarding the product of the roots. We see that we can cancel a negative from both sides to get \[rs(r+s) = b\] \[(r+4)(s-3)(r+s+1)=b + 240.\] Subtracting the first equation from the second equation gives us $(r+4)(s-3)(r+s+1) - rs(r+s) = 240$.

Expanding, simplifying, substituting $s = \frac{5r+13}{2}$, and simplifying some more yields the simple quadratic $r^2 + 4r - 5 = 0$, so $r = -5, 1$. Then $s = -6, 9$.

Finally, we substitute back into $b=rs(r+s)$ to get $b = (-5)(-6)(-5-6) = -330$, or $b = (1)(9)(1 + 9) = 90$.

The answer is $|-330|+|90| = \boxed{420}$.

Solution 4

By Vieta's, we know that the sum of roots of $p(x)$ is $0$. Therefore, the roots of $p$ are $r, s, -r-s$. By similar reasoning, the roots of $q(x)$ are $r + 4, s - 3, -r - s - 1$. Thus, $p(x) = (x - r)(x - s)(x + r + s)$ and $q(x) = (x - r - 4)(x - s + 3)(x + r + s + 1)$.

Since $p(x)$ and $q(x)$ have the same coefficient for $x$, we can go ahead and match those up to get \begin{align*} rs - r(r + s) - s(r + s) &= (r + 4)(s - 3) - (r + 4)(r + s + 1) - (s - 3)(r + s + 1) \\ 0 &= -13 - 5r + 2s \\ s &= \frac{5r + 13}{2} \end{align*}

At this point, we can go ahead and compare the constant term in $p(x)$ and $q(x)$. Doing so is certainly valid, but we can actually do this another way. Notice that $p(s) = 0$. Therefore, $q(s) = 240$. If we plug that into our expression, we get that \begin{align*} q(s) &= 3(s - r - 4)(r + 2s + 1) \\ 240 &= 3(s - r - 4)(r + 2s + 1) \\ 240 &= 3\left( \frac{3r + 5}{2} \right)(6r + 14) \\ 80 &= (3r + 5)(3r + 7) \\ 0 &= r^2 + 4r - 5 \end{align*} This tells us that $(r, s) = (1, 9)$ or $(-5, -6)$. Since $-b$ is the product of the roots, we have that the two possibilities are $1 \cdot 9 \cdot (-10) = -90$ and $(-5)(-6)11 = 330$. Adding the absolute values of these gives us $\boxed{420}$.

See also

2014 AIME II (Problems • Answer Key • Resources)
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