Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2024 AMC 10A Problems/Problem 11"

(→‎Solution 2)
 
(13 intermediate revisions by 8 users not shown)
Line 5: Line 5:
 
<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}</math> Infinitely many
 
<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}</math> Infinitely many
  
−
== Solution ==
+
== Solution 1 ==
  
 
Note that <math>m</math> is a nonnegative integer.
 
Note that <math>m</math> is a nonnegative integer.
Line 12: Line 12:
 
It is clear that <math>n+m\geq n-m,</math> so <math>(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49).</math> Each ordered pair <math>(n+m,n-m)</math> gives one ordered pair <math>(m,n),</math> so there are <math>\boxed{\textbf{(D)}~4}</math> such ordered pairs <math>(m,n).</math>
 
It is clear that <math>n+m\geq n-m,</math> so <math>(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49).</math> Each ordered pair <math>(n+m,n-m)</math> gives one ordered pair <math>(m,n),</math> so there are <math>\boxed{\textbf{(D)}~4}</math> such ordered pairs <math>(m,n).</math>
  
−
== Problem ==
+
<u><b>Remark</b></u>
−
How many ordered pairs of integers <math>(m, n)</math> satisfy <math>\sqrt{n^2 - 49} = m</math>?
 
  
−
<math>\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}</math> Infinitely many
+
From <math>(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49),</math> we get <math>(m,n)=(24,25),(0,7),(0,-7),(24,-25),</math> respectively.
 +
 
 +
~MRENTHUSIASM
  
 
== Solution 2 ==
 
== Solution 2 ==
−
Squaring both sides of the given equation gives <cmath>n^2-49=m^2\rightarrow n^2-m^2=49\rightarrow (n+m)(n-m)=49</cmath>. Splitting <math>49</math> into its factors (keep in mind it doesn't ask for positive integers, so the factors can be double negative, too) gives six cases:
+
Squaring both sides of the given equation gives <cmath>n^2-49=m^2\rightarrow n^2-m^2=49\rightarrow (n+m)(n-m)=49</cmath> Splitting <math>49</math> into its factors (keep in mind it doesn't ask for positive integers, so the factors can be double negative, too) gives six cases:
 +
 
 +
<math>(1\cdot49)</math>
  
−
<math>1\cdot49</math>
+
<math>(7\cdot7)</math>
  
−
<math>7\cdot7</math>
+
<math>(49\cdot1)</math>
  
−
<math>49\cdot1</math>
+
<math>(-1\cdot -49)</math>
  
−
<math>-1\cdot -49</math>
+
<math>(-7\cdot -7)</math>
  
−
<math>-7\cdot -7</math>
+
<math>(-49\cdot -1)</math>.
  
−
<math>-49\cdot -1</math>.
 
 
Note that the square root in the problem doesn't have <math>\pm</math> with it. Therefore, if there are two solutions, <math>(n,m)</math> and <math>(n,-m)</math>, then these together are to be counted as one solution.
 
Note that the square root in the problem doesn't have <math>\pm</math> with it. Therefore, if there are two solutions, <math>(n,m)</math> and <math>(n,-m)</math>, then these together are to be counted as one solution.
 
The solutions expressed as <math>(n,m)</math> are:
 
The solutions expressed as <math>(n,m)</math> are:
 +
 
<math>(25,24)</math>
 
<math>(25,24)</math>
 +
 
<math>(25,-24)</math>
 
<math>(25,-24)</math>
 +
 
<math>(7,0)</math>
 
<math>(7,0)</math>
 +
 
<math>(-7,0)</math>
 
<math>(-7,0)</math>
 +
 
<math>(-25,24)</math>
 
<math>(-25,24)</math>
 +
 
<math>(-25,-24)</math>.
 
<math>(-25,-24)</math>.
 +
 
<math>(25,24)</math> and <math>(25,-24)</math> are to be counted as one, same for <math>(-25,24)</math> and <math>(-25,-24)</math>. Therefore, the solution is <math>\boxed{\text{(D) }4}</math> ~Tacos_are_yummy_1
 
<math>(25,24)</math> and <math>(25,-24)</math> are to be counted as one, same for <math>(-25,24)</math> and <math>(-25,-24)</math>. Therefore, the solution is <math>\boxed{\text{(D) }4}</math> ~Tacos_are_yummy_1
 +
 +
== Solution 3 (Crazy Rush) ==
 +
From looking at the problem, it's obvious that <math>(\pm7,0)</math> are already solutions. From squaring and rearranging, we get <cmath>n^2-m^2=49.</cmath>We know that the difference between two consecutive squares is always odd, and for each pair of increasing consecutive squares, the difference starts from <math>3</math> and increases by <math>2</math> each time. This means that there is an existing pair, <math>(\pm n,m)</math> of consecutive squares that will satisfy this equation.
 +
 +
Also note that the answer cannot be infinity because the difference between two squares will increase as the two squares get higher, consecutive or not.
 +
 +
Therefore, the solutions <math>(\pm7,0)</math> and <math>(\pm n,m)</math> where <math>n</math> and <math>m</math> are consecutive that have a square difference of <math>49</math>, give the answer of <math>\boxed{\text{(D) }4}</math> ~Tacos_are_yummy_1
 +
 +
Minor Note: Looking at the problem, it is obvious of -7 and 7 as solutions and you can eliminate 1,3,and infinite. You can assume that there is more than 2 solutions because when determining square expressions, these problems usually always have more than an obvious solution.
 +
~breakingbread
 +
 +
==Solution 4 (Quick)==
 +
Clearly, <math>(7,0)</math> and <math>(-7,0)</math> are solutions.
 +
Notice that <math>49 = 7^2</math>. Remembering our Pythagorean triples, we realize <math>(25, 24)</math> is a solution as well, and by extension, so is <math>(-25, 24)</math>. Since 7 is not part of any other Pythagorean triple, we can conclude the answer is  <math>\boxed{\text{(D) }4}</math>.
 +
~i_am_suk_at_math_2
 +
 +
==Solution 5 (idk why someone would ever use this)==
 +
 +
Since in <math>n^2-49</math> there are no restrictions of the sign of <math>n</math>, we know that the solution will come in pairs, with negative and positive <math>n</math>. Therefore, the answer will not be <math>1</math> or <math>3</math>. We can further confirm that it is not <math>1</math>, since the problem would probably ask for an operation of <math>n</math> and <math>m</math> instead of the number of solutions. We can also guess it is not <math>2</math>, since then the problem would ask for the sum of one of the variables in the solutions. It seems improbable that over the integers there would be infinite answers. So the answer is <math>\boxed{\textbf{(D) }4}</math>
 +
 +
~megaboy6679
 +
 +
(do not ever use this)
 +
 +
== Video Solution by Pi Academy ==
 +
 +
https://youtu.be/ABkKz0gS1MU?si=ZQBgDMRaJmMPSSMM
 +
 +
==Video Solution 1 by Power Solve ==
 +
 +
https://youtu.be/G91MyH1CycE
 +
 +
== Video Solution by Daily Dose of Math ==
 +
 +
https://youtu.be/Ne1qOZNRxfY
 +
 +
~Thesmartgreekmathdude
 +
 +
==Video Solution by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=6SQ74nt3ynw
 +
==See also==
 +
{{AMC10 box|year=2024|ab=A|num-b=10|num-a=12}}
 +
{{MAA Notice}}

Latest revision as of 21:46, 13 November 2024

Problem

How many ordered pairs of integers $(m, n)$ satisfy $\sqrt{n^2 - 49} = m$?

$\textbf{(A)}~1\qquad\textbf{(B)}~2\qquad\textbf{(C)}~3\qquad\textbf{(D)}~4\qquad\textbf{(E)}$ Infinitely many

Solution 1

Note that $m$ is a nonnegative integer.

We square, rearrange, and apply the difference of squares formula to the given equation: \[(n+m)(n-m)=49.\] It is clear that $n+m\geq n-m,$ so $(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49).$ Each ordered pair $(n+m,n-m)$ gives one ordered pair $(m,n),$ so there are $\boxed{\textbf{(D)}~4}$ such ordered pairs $(m,n).$

Remark

From $(n+m,n-m)=(49,1),(7,7),(-7,-7),(-1,-49),$ we get $(m,n)=(24,25),(0,7),(0,-7),(24,-25),$ respectively.

~MRENTHUSIASM

Solution 2

Squaring both sides of the given equation gives \[n^2-49=m^2\rightarrow n^2-m^2=49\rightarrow (n+m)(n-m)=49\] Splitting $49$ into its factors (keep in mind it doesn't ask for positive integers, so the factors can be double negative, too) gives six cases:

$(1\cdot49)$

$(7\cdot7)$

$(49\cdot1)$

$(-1\cdot -49)$

$(-7\cdot -7)$

$(-49\cdot -1)$.

Note that the square root in the problem doesn't have $\pm$ with it. Therefore, if there are two solutions, $(n,m)$ and $(n,-m)$, then these together are to be counted as one solution. The solutions expressed as $(n,m)$ are:

$(25,24)$

$(25,-24)$

$(7,0)$

$(-7,0)$

$(-25,24)$

$(-25,-24)$.

$(25,24)$ and $(25,-24)$ are to be counted as one, same for $(-25,24)$ and $(-25,-24)$. Therefore, the solution is $\boxed{\text{(D) }4}$ ~Tacos_are_yummy_1

Solution 3 (Crazy Rush)

From looking at the problem, it's obvious that $(\pm7,0)$ are already solutions. From squaring and rearranging, we get \[n^2-m^2=49.\]We know that the difference between two consecutive squares is always odd, and for each pair of increasing consecutive squares, the difference starts from $3$ and increases by $2$ each time. This means that there is an existing pair, $(\pm n,m)$ of consecutive squares that will satisfy this equation.

Also note that the answer cannot be infinity because the difference between two squares will increase as the two squares get higher, consecutive or not.

Therefore, the solutions $(\pm7,0)$ and $(\pm n,m)$ where $n$ and $m$ are consecutive that have a square difference of $49$, give the answer of $\boxed{\text{(D) }4}$ ~Tacos_are_yummy_1

Minor Note: Looking at the problem, it is obvious of -7 and 7 as solutions and you can eliminate 1,3,and infinite. You can assume that there is more than 2 solutions because when determining square expressions, these problems usually always have more than an obvious solution. ~breakingbread

Solution 4 (Quick)

Clearly, $(7,0)$ and $(-7,0)$ are solutions. Notice that $49 = 7^2$. Remembering our Pythagorean triples, we realize $(25, 24)$ is a solution as well, and by extension, so is $(-25, 24)$. Since 7 is not part of any other Pythagorean triple, we can conclude the answer is $\boxed{\text{(D) }4}$. ~i_am_suk_at_math_2

Solution 5 (idk why someone would ever use this)

Since in $n^2-49$ there are no restrictions of the sign of $n$, we know that the solution will come in pairs, with negative and positive $n$. Therefore, the answer will not be $1$ or $3$. We can further confirm that it is not $1$, since the problem would probably ask for an operation of $n$ and $m$ instead of the number of solutions. We can also guess it is not $2$, since then the problem would ask for the sum of one of the variables in the solutions. It seems improbable that over the integers there would be infinite answers. So the answer is $\boxed{\textbf{(D) }4}$

~megaboy6679

(do not ever use this)

Video Solution by Pi Academy

https://youtu.be/ABkKz0gS1MU?si=ZQBgDMRaJmMPSSMM

Video Solution 1 by Power Solve

https://youtu.be/G91MyH1CycE

Video Solution by Daily Dose of Math

https://youtu.be/Ne1qOZNRxfY

~Thesmartgreekmathdude

Video Solution by SpreadTheMathLove

https://www.youtube.com/watch?v=6SQ74nt3ynw

See also

2024 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2024_AMC_10A_Problems/Problem_11&oldid=233381"