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Difference between revisions of "2023 AMC 10B Problems/Problem 14"

(→‎See also)
m (i had a bad day that day ok my writing was tripping)
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== Solution 1 ==
 
== Solution 1 ==
  
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Clearly, <math>m=0,n=0</math> is 1 solution. However there are definitely more, so we apply [https://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick Simon's Favorite Factoring Expression] to get this:
+
Clearly, <math>m=0,n=0</math> is one of the solutions. However, we can be quite sure that there are more, so we apply [https://artofproblemsolving.com/wiki/index.php/Simon%27s_Favorite_Factoring_Trick Simon's Favorite Factoring Trick] to get the following:  
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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m^2+mn+n^2 +mn &= m^2n^2 +mn\\
 
m^2+mn+n^2 +mn &= m^2n^2 +mn\\
 
(m+n)^2 &= m^2n^2 +mn\\
 
(m+n)^2 &= m^2n^2 +mn\\
−
(m+n)^2 &= mn(mn+1)\\
+
(m+n)^2 &= mn(mn+1).\\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
  
−
This basically say that the product of two consecutive numbers <math>mn,mn+1</math> must be a perfect square which is practically impossible except <math>mn=0</math> or <math>mn+1=0</math>.
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Essentially, this says that the product of two consecutive numbers <math>mn,mn+1</math> must be a perfect square. This is practically impossible except <math>mn=0</math> or <math>mn+1=0</math>.  
 
<math>mn=0</math> gives <math>(0,0)</math>.
 
<math>mn=0</math> gives <math>(0,0)</math>.
 
<math>mn=-1</math> gives <math>(1,-1), (-1,1)</math>. Answer: <math>\boxed{\textbf{(C)}3}.</math>
 
<math>mn=-1</math> gives <math>(1,-1), (-1,1)</math>. Answer: <math>\boxed{\textbf{(C)}3}.</math>

Revision as of 21:25, 22 March 2024

Problem

How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$?

$\textbf{(A) }7\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }5$

Solution 1

Clearly, $m=0,n=0$ is one of the solutions. However, we can be quite sure that there are more, so we apply Simon's Favorite Factoring Trick to get the following:

\begin{align*} m^2+mn+n^2 &= m^2n^2\\ m^2+mn+n^2 +mn &= m^2n^2 +mn\\ (m+n)^2 &= m^2n^2 +mn\\ (m+n)^2 &= mn(mn+1).\\ \end{align*}

Essentially, this says that the product of two consecutive numbers $mn,mn+1$ must be a perfect square. This is practically impossible except $mn=0$ or $mn+1=0$. $mn=0$ gives $(0,0)$. $mn=-1$ gives $(1,-1), (-1,1)$. Answer: $\boxed{\textbf{(C)}3}.$

~Technodoggo ~minor edits by lucaswujc

Solution 2

Case 1: $mn = 0$.

In this case, $m = n = 0$.

Case 2: $mn \neq 0$.

Denote $k = {\rm gcd} \left( m, n \right)$. Denote $m = k u$ and $n = k v$. Thus, ${\rm gcd} \left( u, v \right) = 1$.

Thus, the equation given in this problem can be written as \[ u^2 + uv + v^2 = k^2 u^2 v^2 . \]

Modulo $u$, we have $v^2 \equiv 0 \pmod{u}$. Because $\left( u, v \right) = 1$, we must have $|u| = |v| = 1$. Plugging this into the above equation, we get $2 + uv = k^2$. Thus, we must have $uv = -1$ and $k = 1$.

Thus, there are two solutions in this case: $\left( m , n \right) = \left( 1, -1 \right)$ and $\left( m , n \right) = \left( -1, 1 \right)$.

Putting all cases together, the total number of solutions is $\boxed{\textbf{(C) 3}}$.

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

Solution 3 (Discriminant)

We can move all terms to one side and wrote the equation as a quadratic in terms of $n$ to get \[(1-m^2)n^2+(m)n+(m^2)=0.\] The discriminant of this quadratic is \[\Delta = m^2-4(1-m^2)(m^2)=m^2(4m^2-3).\] For $n$ to be an integer, we must have $m^2(4m^2-3)$ be a perfect square. Thus, either $4m^2-3$ is a perfect square or $m^2 = 0$ and $m = 0$. The first case gives $m=-1,1$, which result in the equations $-n+1=0$ and $n-1=0$, for a total of two pairs: $(-1,1)$ and $(1,-1)$. The second case gives the equation $n^2=0$, so it's only pair is $(0,0)$. In total, the total number of solutions is $\boxed{\textbf{(C) 3}}$.

~A_MatheMagician

Solution 4 (Nice Substitution)

Let $x=m+n, y=mn$ then \[x^2-y=y^2\] Completing the square then gives \[4x^2+1=(2y+1)^2\] Since the RHS is a square, clearly the only solutions are $x=0,y=0$ and $x=0,y=-1$. The first gives $(0,0)$ while the second gives $(-1,1)$ and $(1,-1)$ by solving it as a quadratic with roots $m$ and $n$. Thus there are $\boxed{\textbf{(C) 3}}$ solutions.

~ Grolarbear

Video Solution by OmegaLearn

https://youtu.be/5a5caco_YTo

Video Solution

https://youtu.be/Dh1lDI1fHrw

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


See also

2023 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

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