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Difference between revisions of "2023 AIME I Problems/Problem 9"

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Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math>
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==Problem==
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are integers in <math>{−20, −10, −18, . . . , 18, 19, 20}</math>, such that there is a unique integer
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Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c,</math> where <math>a, b,</math> and <math>c</math> are integers in <math>\{-20,-19,-18,\ldots,18,19,20\},</math> such that there is a unique integer <math>m \not= 2</math> with <math>p(m) = p(2).</math>
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<math>m \neq 2</math> with <math>p(m) = p(2)</math>.
 
  
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Find the number of cubic polynomials <math>p(x) = x^3 + ax^2 + bx + c</math>, where <math>a</math>, <math>b</math>, and <math>c</math>
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==Solution 1 (bash) ==
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are integers in <math>{-20, -10, -18, . . . , 18, 19, 20}</math>, such that there is a unique integer
 
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<math>m \neq 2</math> with <math>p(m) = p(2)</math>.
 
  
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==Solution==
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Plugging <math>2</math> and <math>m</math> into <math>P(x)</math> and equating them, we get <math>8+4a+2b+c = m^3+am^2+bm+c</math>. Rearranging, we have <cmath>(m^3-8) + (m^2 - 4)a + (m-2)b = 0.</cmath> Note that the value of <math>c</math> won't matter as it can be anything in the provided range, giving a total of <math>41</math> possible choices for <math>c.</math> So what we just need to do is to just find the number of ordered pairs <math>(a, b)</math> that work, and multiply it by <math>41.</math>
 +
We can start by first dividing both sides by <math>m-2.</math> (Note that this is valid since <math>m\neq2:</math> <cmath>m^2 + 2m + 4 + (m+2)a + b = 0.</cmath> We can rearrange this so it is a quadratic in <math>m</math>: <cmath>m^2 + (a+2)m + (4 + 2a + b) = 0.</cmath> Remember that <math>m</math> has to be unique and not equal to <math>2.</math> We can split this into two cases: case <math>1</math> being that <math>m</math> has exactly one solution, and it isn't equal to <math>2</math>; case <math>2</math> being that <math>m</math> has two solutions, one being equal to <math>2,</math> but the other is a unique solution not equal to <math>2.</math>
  
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Define <math>q \left( x \right) = p \left( x \right) - p \left( 2 \right)</math>.
 
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Hence, for <math>q \left( x \right)</math>, beyond having a root 2, it has a unique integer root that is not equal to 2.
 
  
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We have
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<math>\textbf{Case 1:}</math>
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<cmath>
 
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\begin{align*}
 
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q \left( x \right) & = p \left( x \right) - p \left( 2 \right) \\
 
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& = \left( x - 2 \right)
 
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\left( x^2 + \left( 2 + a \right) x + 4 + 2a + b \right) .
 
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\end{align*}
 
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</cmath>
 
  
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Thus, the polynomial <math>x^2 + \left( 2 + a \right) x + 4 + 2a + b</math> has a unique integer root and it is not equal to 2.
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There is exactly one solution for <math>m,</math> and that solution is not <math>2.</math> This means that the discriminant of the quadratic equation is <math>0,</math> using that, we have <math>(a+2)^2 = 4(4 + 2a + b),</math> rearranging in a neat way, we have <cmath>(a-2)^2 = 4(4 + b)\Longrightarrow a = 2\pm2\sqrt{4+b}.</cmath> Using the fact that <math>4+b</math> must be a perfect square, we can easily see that the values for <math>b</math> can be <math>-4, -3, 0, 5,</math> and <math>12.</math> Also since it's a "<math>\pm</math>" there will usually be <math>2</math> solutions for <math>a</math> for each value of <math>b.</math> The two exceptions for this would be if <math>b = -4</math> and <math>b = 12.</math> For <math>b=-4</math> because it would be a <math>\pm0,</math> which only gives one solution, instead of two. And for <math>b=12</math> because then <math>a = -6</math> and the solution for <math>m</math> would equal to <math>2,</math> and we don't want this. (We can know this by putting the solutions back into the quadratic formula).  
  
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Following from Vieta' formula, the sum of two roots of this polynomial is <math>- 2 - a</math>.
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So we have <math>5</math> solutions for <math>b,</math> each of which give <math>2</math> values for <math>a,</math> except for <math>2,</math> which only give one. So in total, there are <math>5*2 - 2 = 8</math> ordered pairs of <math>(a,b)</math> in this case.
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Because <math>a</math> is an integer, if a root is an integer, the other root is also an integer.
 
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Therefore, the only way to have a unique integer root is that the determinant of this polynomial is 0.
 
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Thus,
 
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<cmath>
 
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\[
 
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\left( 2 + a \right)^2 = 4 \left( 4 + 2a + b \right) . \hspace{1cm} (1)
 
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\]
 
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</cmath>
 
  
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In addition, because two identical roots are not 2, we have
 
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<cmath>
 
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\[
 
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2 + a \neq - 4 .
 
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\]
 
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</cmath>
 
  
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Equation (1) can be reorganized as
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<math>\textbf{Case 2:}</math>
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<cmath>
 
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\[
 
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4 b = \left( a - 2 \right)^2 - 16 .  \hspace{1cm} (2)
 
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\]
 
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</cmath>
 
  
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Thus, <math>2 | a</math>. Denote <math>d = \frac{a-2}{2}</math>.
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<math>m</math> has two solutions, but exactly one of them isn't equal to <math>2.</math> This ensures that <math>1</math> of the solutions is equal to <math>2.</math>
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Thus, (2) can be written as
 
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<cmath>
 
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\[
 
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b = d^2 - 4 . \hspace{1cm} (3)
 
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\]
 
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</cmath>
 
  
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Because <math>a \in \left\{ -20, -19, -18, \cdots , 18, 19, 20 \right\}</math>, <math>2 | a</math>, and <math>2 + a \neq -4</math>, we have <math>d \in \left\{ - 11, - 10, \cdots, 9 \right\} \backslash \left\{ 4 \right\}</math>.
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Let <math>r</math> be the other value of <math>m</math> that isn't <math>2.</math> By Vieta:
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<cmath>\begin{align*}
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r+2 &= -a-2\\
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2r &= 4+2a+b.
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\end{align*}</cmath> From the first equation, we subtract both sides by <math>2</math> and double both sides to get <math>2r = -2a - 8</math> which also equals to <math>4+2a+b</math> from the second equation. Equating both, we have <math>4a + b + 12 = 0.</math> We can easily count that there would be <math>11</math> ordered pairs <math>(a,b)</math> that satisfy that.
  
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Therefore, we have the following feasible solutions for <math>\left( b, d \right)</math>: <math>\left( -4 , 0 \right)</math>, <math>\left( -3 , \pm 1 \right)</math>, <math>\left( 0 , \pm 2 \right)</math>, <math>\left( 5, \pm 3 \right)</math>, <math>\left( 12 , 4 \right)</math>.
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However, there's an outlier case in which <math>r</math> happens to also equal to <math>2,</math> and we don't want that. We can reverse engineer and find out that <math>r=2</math> when <math>(a,b) = (-6, 12),</math> which we overcounted. So we subtract by one and we conclude that there are <math>10</math> ordered pairs of <math>(a,b)</math> that satisfy this case.
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Thus, the total number of <math>\left( b, d \right)</math> is 8.
 
  
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Because <math>c</math> can take any value from <math>\left\{ -20, -19, -18, \cdots , 18, 19, 20 \right\}</math>, the number of feasible <math>c</math> is 41.
 
  
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Therefore, the number of <math>\left( a, b, c \right)</math> is <math>8 \cdot 41 = \boxed{\textbf{(328) }}</math>.
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This all shows that there are a total of <math>8+10 = 18</math> amount of ordered pairs <math>(a,b).</math> Multiplying this by <math>41</math> (the amount of values for <math>c</math>) we get <math>18\cdot41=\boxed{738}</math> as our final answer.
  
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~s214425
  
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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==Solution 2 (factor the difference)==
  
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==Solution==
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<math>p(x)-p(2)</math> is a cubic with at least two integral real roots, therefore it has three real roots, which are all integers.
  
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It can be easily noticed that <math>c</math> is independent of the condition <math>P(m) = P(2)</math>, and can thus safely take all <math>41</math> possible values between <math>-20</math> and <math>20</math>.
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There are exactly two distinct roots, so either <math>p(x)=p(2)+(x-2)^2(x-m)</math> or <math>p(x)=p(2)+(x-2)(x-m)^2</math>, with <math>m\neq 2</math>.
  
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There are two possible ways for <math>m\ne2</math> to be the only integer satisfying <math>P(m) = P(2)</math>: <math>P</math> has a double root at <math>2</math> or a double root at <math>m</math>.
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In the first case <math>p(x)=x^3-(4+m)x^2+(4+4m)x-4m+p(2)</math>, with <math>|4+4m|\leq 20</math> (which entails <math>|4+m|\leq 20</math>), so <math>m</math> can be <math>-6,-5,-4,-3,-2,-1,0,1, (\textbf{not 2}!), 3,4</math> and <math>-4m+p(2)</math> can be any integer from <math>-20</math> to <math>20</math>, giving <math>410</math> polynomials (since the coefficients are given by linear functions of <math>m</math> and thus are distinct).  
  
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Case 1: <math>P</math> has a double root at <math>2</math>:
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In the second case <math>p(x)=x^3-(2+2m)x^2+(4m+m^2)x-2m^2+p(2)</math>, and <math>m</math> can be <math>-6,-5,-4,-3,-2,-1,0,1</math> and <math>-4m+p(2)</math> can be any integer from <math>-20</math> to <math>20</math>, giving <math>328</math> polynomials.
  
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In this case, <math>\frac{dP}{dx}(2) = 0</math>, or <math>12 + 4a + b = 0</math>. Thus <math>a</math> ranges from <math>-8</math> to <math>2</math>. One of these values, <math>(a,b) = (-6,-12)</math> corresponds to a triple root at <math>2</math>, which means <math>m=2</math>. Thus there are <math>10</math> possible values of <math>(a,b)</math>. (It can be verified that <math>m</math> is an integer).
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The total is <math>\boxed{738}</math>.
  
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Case 2: <math>P</math> has a double root at <math>m</math>:
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~EVIN-
  
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See the above solution. This yields <math>8</math> possible combinations of <math>a</math> and <math>b</math>.
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==Video Solution==
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https://youtu.be/-Asb_5nTgSg
  
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Thus, in total we have <math>41*18 = \boxed{738}</math> combinations of <math>(a,b,c)</math>.
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~MathProblemSolvingSkills.com
  
  
  
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-Alex_Z
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==See also==
 +
{{AIME box|year=2023|num-b=8|num-a=10|n=I}}
 +
 
 +
[[Category:Intermediate Algebra Problems]]
 +
{{MAA Notice}}

Latest revision as of 19:40, 27 December 2023

Problem

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c,$ where $a, b,$ and $c$ are integers in $\{-20,-19,-18,\ldots,18,19,20\},$ such that there is a unique integer $m \not= 2$ with $p(m) = p(2).$

Solution 1 (bash)

Plugging $2$ and $m$ into $P(x)$ and equating them, we get $8+4a+2b+c = m^3+am^2+bm+c$. Rearranging, we have \[(m^3-8) + (m^2 - 4)a + (m-2)b = 0.\] Note that the value of $c$ won't matter as it can be anything in the provided range, giving a total of $41$ possible choices for $c.$ So what we just need to do is to just find the number of ordered pairs $(a, b)$ that work, and multiply it by $41.$ We can start by first dividing both sides by $m-2.$ (Note that this is valid since $m\neq2:$ \[m^2 + 2m + 4 + (m+2)a + b = 0.\] We can rearrange this so it is a quadratic in $m$: \[m^2 + (a+2)m + (4 + 2a + b) = 0.\] Remember that $m$ has to be unique and not equal to $2.$ We can split this into two cases: case $1$ being that $m$ has exactly one solution, and it isn't equal to $2$; case $2$ being that $m$ has two solutions, one being equal to $2,$ but the other is a unique solution not equal to $2.$


$\textbf{Case 1:}$

There is exactly one solution for $m,$ and that solution is not $2.$ This means that the discriminant of the quadratic equation is $0,$ using that, we have $(a+2)^2 = 4(4 + 2a + b),$ rearranging in a neat way, we have \[(a-2)^2 = 4(4 + b)\Longrightarrow a = 2\pm2\sqrt{4+b}.\] Using the fact that $4+b$ must be a perfect square, we can easily see that the values for $b$ can be $-4, -3, 0, 5,$ and $12.$ Also since it's a "$\pm$" there will usually be $2$ solutions for $a$ for each value of $b.$ The two exceptions for this would be if $b = -4$ and $b = 12.$ For $b=-4$ because it would be a $\pm0,$ which only gives one solution, instead of two. And for $b=12$ because then $a = -6$ and the solution for $m$ would equal to $2,$ and we don't want this. (We can know this by putting the solutions back into the quadratic formula).

So we have $5$ solutions for $b,$ each of which give $2$ values for $a,$ except for $2,$ which only give one. So in total, there are $5*2 - 2 = 8$ ordered pairs of $(a,b)$ in this case.


$\textbf{Case 2:}$

$m$ has two solutions, but exactly one of them isn't equal to $2.$ This ensures that $1$ of the solutions is equal to $2.$

Let $r$ be the other value of $m$ that isn't $2.$ By Vieta: \begin{align*} r+2 &= -a-2\\ 2r &= 4+2a+b. \end{align*} From the first equation, we subtract both sides by $2$ and double both sides to get $2r = -2a - 8$ which also equals to $4+2a+b$ from the second equation. Equating both, we have $4a + b + 12 = 0.$ We can easily count that there would be $11$ ordered pairs $(a,b)$ that satisfy that.

However, there's an outlier case in which $r$ happens to also equal to $2,$ and we don't want that. We can reverse engineer and find out that $r=2$ when $(a,b) = (-6, 12),$ which we overcounted. So we subtract by one and we conclude that there are $10$ ordered pairs of $(a,b)$ that satisfy this case.


This all shows that there are a total of $8+10 = 18$ amount of ordered pairs $(a,b).$ Multiplying this by $41$ (the amount of values for $c$) we get $18\cdot41=\boxed{738}$ as our final answer.

~s214425

Solution 2 (factor the difference)

$p(x)-p(2)$ is a cubic with at least two integral real roots, therefore it has three real roots, which are all integers.

There are exactly two distinct roots, so either $p(x)=p(2)+(x-2)^2(x-m)$ or $p(x)=p(2)+(x-2)(x-m)^2$, with $m\neq 2$.

In the first case $p(x)=x^3-(4+m)x^2+(4+4m)x-4m+p(2)$, with $|4+4m|\leq 20$ (which entails $|4+m|\leq 20$), so $m$ can be $-6,-5,-4,-3,-2,-1,0,1, (\textbf{not 2}!), 3,4$ and $-4m+p(2)$ can be any integer from $-20$ to $20$, giving $410$ polynomials (since the coefficients are given by linear functions of $m$ and thus are distinct).

In the second case $p(x)=x^3-(2+2m)x^2+(4m+m^2)x-2m^2+p(2)$, and $m$ can be $-6,-5,-4,-3,-2,-1,0,1$ and $-4m+p(2)$ can be any integer from $-20$ to $20$, giving $328$ polynomials.

The total is $\boxed{738}$.

~EVIN-

Video Solution

https://youtu.be/-Asb_5nTgSg

~MathProblemSolvingSkills.com


See also

2023 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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