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 "2015 AMC 10A Problems/Problem 16"

(→‎Solution 1)
(→‎Problem)
 
(42 intermediate revisions by 16 users not shown)
Line 23: Line 23:
 
Moving terms gives the equation
 
Moving terms gives the equation
  
−
<math>x^2 - y^2 = 3x - 3y.</math>
+
<math>x^2 - y^2 = (x + y)(x - y) = 3x - 3y = 3(x - y).</math>
  
 
Because <math>x \neq y,</math> we can divide both sides of the equation by <math>x - y</math> to yield the equation
 
Because <math>x \neq y,</math> we can divide both sides of the equation by <math>x - y</math> to yield the equation
Line 33: Line 33:
 
<math>x^2 + y^2 = 5 \left( x + y \right) = 5 \left( 3 \right) = \boxed{\textbf{(B) } 15}</math>
 
<math>x^2 + y^2 = 5 \left( x + y \right) = 5 \left( 3 \right) = \boxed{\textbf{(B) } 15}</math>
  
−
==Solution 2 (Algebraic) ==
+
===Solution 2 (Algebraic) ===
  
 
Subtract <math>4</math> from the left hand side of both equations, and use difference of squares to yield the equations
 
Subtract <math>4</math> from the left hand side of both equations, and use difference of squares to yield the equations
Line 63: Line 63:
 
<math>\frac{3^2 + 21 + 6\sqrt{21}}{4} + \frac{3^2 + 21 - 6\sqrt{21}}{4} = \frac{2(3^2 +21)}{4} = \boxed{\textbf{(B) } 15}</math>.
 
<math>\frac{3^2 + 21 + 6\sqrt{21}}{4} + \frac{3^2 + 21 - 6\sqrt{21}}{4} = \frac{2(3^2 +21)}{4} = \boxed{\textbf{(B) } 15}</math>.
  
−
==Solution 3==
+
====Alternative Ending====
 +
Note that <math>x^2+y^2 = (x+y)^2 - 2xy</math>. Using Vieta's, we find that <math>(x+y)^2 = 3^2 = 9</math> and <math>-2xy = -2\cdot 3 = -6</math>. Subtracting them gives <math>\boxed{\textbf{(B) } 15}</math>.
 +
 
 +
~Andrew2019
 +
 
 +
===Solution 3===
 
By graphing the two equations on a piece of graph paper, we can see that the point where they intersect that is not on the line <math>y=x</math> is close to the point <math>(4,-1)</math> (or <math>(-1, 4)</math>). <math>(-1)^2+4^2=17</math>, and the closest answer choice to <math>17</math> is <math>\boxed{\textbf{(B) } 15}</math>.
 
By graphing the two equations on a piece of graph paper, we can see that the point where they intersect that is not on the line <math>y=x</math> is close to the point <math>(4,-1)</math> (or <math>(-1, 4)</math>). <math>(-1)^2+4^2=17</math>, and the closest answer choice to <math>17</math> is <math>\boxed{\textbf{(B) } 15}</math>.
  
−
Note: This is risky, as <math>20</math> could be a viable answer too. Do not use this method unless sure about the answer. Also, very time consuming to graph the equations, do not attempt with time limit.
+
====Note====
 +
This is risky, as <math>20</math> could be a viable answer too. Do not use this method unless you're sure about the answer. In other words, this solution is less reliable than the others, so only use it if you can't do the other methods.
 +
 
 +
===Solution 4 (When you can't algebraically manipulate)===
 +
This, by the way, is a really cheap way of solving the problem but as long as you get an answer, it doesn't matter.
 +
 
 +
Looking at the first given equation, begin searching for solutions. Notice how <math>(4,0)</math> works but when plugged into the
 +
 
 +
second equation, you get <math>8=4</math>. Now, if you decrease the <math>y</math> value by <math>1</math> to obtain <math>(2+\sqrt3,-1)</math>, plugging it into
 +
 
 +
the second equation will yield <math>6+\sqrt3=9</math>. Now, notice how the LHS is now less than the RHS as opposed to what it had
 +
 
 +
been when we plugged <math>(4,0)</math> into the second given equation. We then conclude that there must be a solution between
 +
 
 +
<math>y=0</math> and <math>y=-1</math>, so calculating <math>x^2+y^2</math> of <math>(4,0)</math> and <math>(2+\sqrt3,-1)</math> we obtain
 +
 
 +
<cmath>4^2+0^2=16</cmath> and
 +
<cmath>(2+\sqrt3)^2+(-1)^2=8+4\sqrt3\approx14.9</cmath>
 +
 
 +
Thus we know the answer to be <math>\boxed{\textbf{(B) } 15}</math>.
 +
 
 +
This is a really bad solution, but it works. :/
 +
 
 +
===Solution 5 (Very in depth and reasoned) ===
 +
 
 +
Note: In this solution, I explain my entire thought process and the reason why I did each step, so you can read a more intuitive solution rather than a contrived one.
 +
 
 +
We first notice that each equation has a <math>4</math>, which inspires us to isolate the <math>4</math> on both equations and equate them to see if we can gather any useful information on <math>x</math> and <math>y.</math>
 +
 
 +
Doing this we get <math>(x-2)^2-y = (y-2)^2- x.</math> Now you might be tempted to expand, but notice how we have two perfect squares in our equation: <math>(x-2)^2</math> and <math>(y-2)^2,</math> which motivates us to take advantage of the difference of squares.
 +
For simplicity, we suppose that <math>(x-2) = a</math> and <math>(y-2) = b.</math> Thus our equation is now <math>a^2-y = b^2-x.</math> We manipulate this equation by getting the LHS in a form that lets us do difference of squares, so we can rewrite as <math>a^2-b^2 = -(x-y).</math>
 +
 
 +
Now we can finally use our difference of squares to get <math>(a-b)(a+b) =  -(x-y).</math>
 +
 
 +
Let's now substitute <math>a</math> and <math>b</math> back to get information solely for <math>x</math> and <math>y.</math>
 +
 
 +
Doing this we get, <math>(x-y)(x+y-4) = -(x-y).</math>
 +
 
 +
Now let us divide <math>x-y</math> on both sides and clean up the equation. When we do this, we get that <math>x+y = 3.</math>
 +
 
 +
We got <math>x+y = 3,</math> which means that this is fact is only of use if we can obtain it in another way, so we add the LHS and the RHS of both equations and equate them to make our fact of essence.
 +
 
 +
Doing that, we get <math>x+y +8 = (x-2)^2+ (y-2)^2.</math> Also, when we set this equation up we not only do we make our fact useful, but we also can see that after expanding the RHS, we get <math>x^2+y^2</math> on the RHS too, which makes it even a better move that we added both equations because remember our goal is to find <math>x^2+y^2.</math>
 +
 
 +
Expanding and plugging in <math>x+y = 3</math> we get <math>11 = x^2+y^2-4(x+y)+8.</math> <math>x+y=3,</math> so we can plug that in and we get <math>x^2+y^2 = 15</math> or <math>\boxed{\textbf{(B) } 15}</math> after cleaning up some algebra.
 +
 
 +
~triggod
 +
 
 +
===Solution 6 (Quickest) ===
 +
Subtract the two equations to get <math>(x-y)= (x+y-4)(y-x) \iff x+y=3</math>. Plugging back into any of the original equations yields <math>x^2-3x=3</math>. However, we know <math>x^2+y^2= x^2+(3-x)^2= 2x^2-6x+9</math>. Then, we can just multiply the first by two and add <math>9</math> to get <math>\boxed{\textbf{(B) } 15}</math>
 +
 
 +
==Video Solutions==
 +
=== Video Solution by OmegaLearn ===
 +
https://youtu.be/0W3VmFp55cM?t=15
 +
 
 +
~ pi_is_3.14
 +
 
 +
===Video Solution 2===
 +
https://youtu.be/WsRGBqCJwGA
 +
 
 +
~savannahsolver
  
 
==See Also==
 
==See Also==

Latest revision as of 23:09, 23 September 2024

Problem

If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?

$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30}$

Solutions

Solution 1

Note that we can add the two equations to yield the equation

$x^2 + y^2 - 4x - 4y + 8 = x + y + 8.$

Moving terms gives the equation

$x^2+y^2=5 \left( x + y \right).$

We can also subtract the two equations to yield the equation

$x^2 - y^2 - 4x +4y = y - x.$

Moving terms gives the equation

$x^2 - y^2 = (x + y)(x - y) = 3x - 3y = 3(x - y).$

Because $x \neq y,$ we can divide both sides of the equation by $x - y$ to yield the equation

$x + y = 3.$

Substituting this into the equation for $x^2 + y^2$ that we derived earlier gives

$x^2 + y^2 = 5 \left( x + y \right) = 5 \left( 3 \right) = \boxed{\textbf{(B) } 15}$

Solution 2 (Algebraic)

Subtract $4$ from the left hand side of both equations, and use difference of squares to yield the equations

$x = y(y-4)$ and $y = x(x-4)$.

It may save some time to find two solutions, $(0, 0)$ and $(5, 5)$, at this point. However, $x = y$ in these solutions.


Substitute $y = x(x-4)$ into $x = y(y-4)$.


This gives the equation

$x = x(x-4)(x^2-4x-4)$

which can be simplified to

$x(x^3 - 8x^2 +12x + 15) = 0$.

Knowing $x = 0$ and $x = 5$ are solutions is now helpful, as you divide both sides by $x(x-5)$. This can also be done using polynomial division to find $x = 5$ as a factor. This gives

$x^2 - 3x -3 = 0$.

Because the two equations $x = y(y-4)$ and $y = x(x-4)$ are symmetric, the $x$ and $y$ values are the roots of the equation, which are $x = \frac{3 + \sqrt{21}}{2}$ and $x = \frac{3 - \sqrt{21}}{2}$.

Squaring these and adding them together gives

$\frac{3^2 + 21 + 6\sqrt{21}}{4} + \frac{3^2 + 21 - 6\sqrt{21}}{4} = \frac{2(3^2 +21)}{4} = \boxed{\textbf{(B) } 15}$.

Alternative Ending

Note that $x^2+y^2 = (x+y)^2 - 2xy$. Using Vieta's, we find that $(x+y)^2 = 3^2 = 9$ and $-2xy = -2\cdot 3 = -6$. Subtracting them gives $\boxed{\textbf{(B) } 15}$.

~Andrew2019

Solution 3

By graphing the two equations on a piece of graph paper, we can see that the point where they intersect that is not on the line $y=x$ is close to the point $(4,-1)$ (or $(-1, 4)$). $(-1)^2+4^2=17$, and the closest answer choice to $17$ is $\boxed{\textbf{(B) } 15}$.

Note

This is risky, as $20$ could be a viable answer too. Do not use this method unless you're sure about the answer. In other words, this solution is less reliable than the others, so only use it if you can't do the other methods.

Solution 4 (When you can't algebraically manipulate)

This, by the way, is a really cheap way of solving the problem but as long as you get an answer, it doesn't matter.

Looking at the first given equation, begin searching for solutions. Notice how $(4,0)$ works but when plugged into the

second equation, you get $8=4$. Now, if you decrease the $y$ value by $1$ to obtain $(2+\sqrt3,-1)$, plugging it into

the second equation will yield $6+\sqrt3=9$. Now, notice how the LHS is now less than the RHS as opposed to what it had

been when we plugged $(4,0)$ into the second given equation. We then conclude that there must be a solution between

$y=0$ and $y=-1$, so calculating $x^2+y^2$ of $(4,0)$ and $(2+\sqrt3,-1)$ we obtain

\[4^2+0^2=16\] and \[(2+\sqrt3)^2+(-1)^2=8+4\sqrt3\approx14.9\]

Thus we know the answer to be $\boxed{\textbf{(B) } 15}$.

This is a really bad solution, but it works. :/

Solution 5 (Very in depth and reasoned)

Note: In this solution, I explain my entire thought process and the reason why I did each step, so you can read a more intuitive solution rather than a contrived one.

We first notice that each equation has a $4$, which inspires us to isolate the $4$ on both equations and equate them to see if we can gather any useful information on $x$ and $y.$

Doing this we get $(x-2)^2-y = (y-2)^2- x.$ Now you might be tempted to expand, but notice how we have two perfect squares in our equation: $(x-2)^2$ and $(y-2)^2,$ which motivates us to take advantage of the difference of squares. For simplicity, we suppose that $(x-2) = a$ and $(y-2) = b.$ Thus our equation is now $a^2-y = b^2-x.$ We manipulate this equation by getting the LHS in a form that lets us do difference of squares, so we can rewrite as $a^2-b^2 = -(x-y).$

Now we can finally use our difference of squares to get $(a-b)(a+b) = -(x-y).$

Let's now substitute $a$ and $b$ back to get information solely for $x$ and $y.$

Doing this we get, $(x-y)(x+y-4) = -(x-y).$

Now let us divide $x-y$ on both sides and clean up the equation. When we do this, we get that $x+y = 3.$

We got $x+y = 3,$ which means that this is fact is only of use if we can obtain it in another way, so we add the LHS and the RHS of both equations and equate them to make our fact of essence.

Doing that, we get $x+y +8 = (x-2)^2+ (y-2)^2.$ Also, when we set this equation up we not only do we make our fact useful, but we also can see that after expanding the RHS, we get $x^2+y^2$ on the RHS too, which makes it even a better move that we added both equations because remember our goal is to find $x^2+y^2.$

Expanding and plugging in $x+y = 3$ we get $11 = x^2+y^2-4(x+y)+8.$ $x+y=3,$ so we can plug that in and we get $x^2+y^2 = 15$ or $\boxed{\textbf{(B) } 15}$ after cleaning up some algebra.

~triggod

Solution 6 (Quickest)

Subtract the two equations to get $(x-y)= (x+y-4)(y-x) \iff x+y=3$. Plugging back into any of the original equations yields $x^2-3x=3$. However, we know $x^2+y^2= x^2+(3-x)^2= 2x^2-6x+9$. Then, we can just multiply the first by two and add $9$ to get $\boxed{\textbf{(B) } 15}$

Video Solutions

Video Solution by OmegaLearn

https://youtu.be/0W3VmFp55cM?t=15

~ pi_is_3.14

Video Solution 2

https://youtu.be/WsRGBqCJwGA

~savannahsolver

See Also

2015 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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=2015_AMC_10A_Problems/Problem_16&oldid=227912"