2023 AIME I Problems/Problem 9

Problem 9

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers in $\{ -20, -19, -18, \dots , 18, 19, 20 \}$ , such that there is a unique integer $m \neq 2$ with $p(m) = p(2).$

Solution

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 $(a-2)^2 = 4(4 + b)\Longrightarrow a = 2\pm2sqrt{4+b}.$ 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 $5*10 - 2 = 8$ 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: $r + 2 = -a - 2$ $2r = 4 + 2a + b.$ 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 1

Plugging $2$ into $P(x)$ , we get $8+4a+2b+c = m^3+am^2+bm+c$ . We can rewrite into $(2-m)(m^2+2m+4+a(2+m)+b)=0$ , where $c$ can be any value in the range. Since $m\neq2, m^2+2m+4+a(2+m)+b$ must be $0$ . The problem also asks for unique integers, meaning $m$ can only be one value for each polynomial, so the discriminant must be $0$ . $m^2+2m+4+a(2+m)+b = m^2+m(2+a)+(2a+b+4)= 0$ , and $(2+a)^2-4(2a+b+4)=0$ . Rewrite to be $a(a-4)=4(b+3)$ . $a$ must be even for $4(b+3)$ to be an integer. $-6<=a<=10$ because $4(20+3) = 92$ . However, plugging in $a=-6, b=12$ result in $m=2$ . There are 8 pairs of $(a,b)$ and 41 integers for $c$ , giving $41\cdot8 = \boxed{328}$ ~chem1kall

Solution 2

Define q $\left( x \right) = p \left( x \right) - p \left( 2 \right)$ . Hence, for $q \left( x \right)$ , beyond having a root 2, it has a unique integer root that is not equal to 2.

We have $q(x)=p(x)-p(2)=(x-2)( (x^2+2x+4) + a(x+2) + b)$ Thus, the polynomial $x^2 + \left( 2 + a \right) x + 4 + 2a + b$ has a unique integer root and it is not equal to 2.

Following from Vieta' formula, the sum of two roots of this polynomial is $- 2 - a$ . Because $a$ is an integer, if a root is an integer, the other root is also an integer. Therefore, the only way to have a unique integer root is that the determinant of this polynomial is 0. Thus, $\left( 2 + a \right)^2 = 4 \left( 4 + 2a + b \right) . \hspace{1cm} (1)$ In addition, because two identical roots are not 2, we have $2 + a \neq - 4 .$ Equation (1) can be reorganized as $4 b = \left( a - 2 \right)^2 - 16 . \hspace{1cm} (2)$ Thus, $2 | a$ . Denote $d = \frac{a-2}{2}$ . Thus, (2) can be written as $b = d^2 - 4 . \hspace{1cm} (3)$ Because $a \in \left\{ -20, -19, -18, \cdots , 18, 19, 20 \right\}$ , $2 | a$ , and $2 + a \neq -4$ , we have $d \in \left\{ - 11, - 10, \cdots, 9 \right\} \backslash \left\{ 4 \right\}$ .

Therefore, we have the following feasible solutions for $\left( b, d \right)$ : $\left( -4 , 0 \right)$ , $\left( -3 , \pm 1 \right)$ , $\left( 0 , \pm 2 \right)$ , $\left( 5, \pm 3 \right)$ , $\left( 12 , 4 \right)$ . Thus, the total number of $\left( b, d \right)$ is 8.

Because $c$ can take any value from $\left\{ -20, -19, -18, \cdots , 18, 19, 20 \right\}$ , the number of feasible $c$ is 41.

Therefore, the number of $\left( a, b, c \right)$ is $8 \cdot 41 = \boxed{\textbf{(328) }}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 3

$p(x)-p(2)$ is a cubic with 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 $p(x)=x^3-(4+m)x^2+(4+4m)x-4m+p(2)$ , so $m$ can be $-6,-5,-4,-3,-2,-1,0,1,3,4$ and $-4m+p(2)$ can be any integer from $-20$ to $20$ , giving $410$ polynomials. In the second case $p(x)=x^3-(1+2m)x^2+(4m+m^2)x-2m^2+p(2)$ , 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-

2023 AIME I (Problems • Answer Key • Resources)
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