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Difference between revisions of "2020 AMC 10B Problems/Problem 9"

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==Problem==
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{{duplicate|[[2020 AMC 10B Problems|2020 AMC 10B #9]] and [[2020 AMC 12B Problems|2020 AMC 12B #8]]}}
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== Problem ==
  
 
How many ordered pairs of integers <math>(x, y)</math> satisfy the equation <cmath>x^{2020}+y^2=2y?</cmath>
 
How many ordered pairs of integers <math>(x, y)</math> satisfy the equation <cmath>x^{2020}+y^2=2y?</cmath>
 +
 
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}</math>
 
<math>\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{infinitely many}</math>
  
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==Solution==
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== Solutions ==
 +
=== Solution 1 ===
 
Rearranging the terms and and completing the square for <math>y</math> yields the result <math>x^{2020}+(y-1)^2=1</math>. Then, notice that <math>x</math> can only be <math>0</math>, <math>1</math> and <math>-1</math> because any value of <math>x^{2020}</math> that is greater than 1 will cause the term <math>(y-1)^2</math> to be less than <math>0</math>, which is impossible as <math>y</math> must be real. Therefore, plugging in the above values for <math>x</math> gives the ordered pairs <math>(0,0)</math>, <math>(1,1)</math>, <math>(-1,1)</math>, and <math>(0,2)</math> gives  a  total  of <math>\boxed{\textbf{(D) }4}</math> ordered pairs.
 
Rearranging the terms and and completing the square for <math>y</math> yields the result <math>x^{2020}+(y-1)^2=1</math>. Then, notice that <math>x</math> can only be <math>0</math>, <math>1</math> and <math>-1</math> because any value of <math>x^{2020}</math> that is greater than 1 will cause the term <math>(y-1)^2</math> to be less than <math>0</math>, which is impossible as <math>y</math> must be real. Therefore, plugging in the above values for <math>x</math> gives the ordered pairs <math>(0,0)</math>, <math>(1,1)</math>, <math>(-1,1)</math>, and <math>(0,2)</math> gives  a  total  of <math>\boxed{\textbf{(D) }4}</math> ordered pairs.
  
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==Video Solution==
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=== Solution 2 ===
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https://youtu.be/OHR_6U686Qg
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Bringing all of the terms to the LHS, we see a quadratic equation <cmath>y^2 - 2y + x^{2020} = 0</cmath> in terms of <math>y</math>. Applying the quadratic formula, we get <cmath>y = \frac{2\pm\sqrt{4-4\cdot1\cdot x^{2020}}}{2}=\frac{2\pm\sqrt{4(1-x^{2020})}}{2}.</cmath> In order for <math>y</math> to be real, which it must be given the stipulation that we are seeking integral answers, we know that the discriminant, <math>4(1-x^{2020})</math> must be nonnegative. Therefore, <cmath>4(1-x^{2020}) \geq 0 \implies x^{2020} \leq 1.</cmath> Here, we see that we must split the inequality into a compound, resulting in <math>-1 \leq x \leq 1</math>.
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 +
The only integers that satisfy this are <math>x \in \{-1,0,1\}</math>. Plugging these values back into the quadratic equation, we see that <math>x = \{-1,1\}</math> both produce a discriminant of <math>0</math>, meaning that there is only 1 solution for <math>y</math>. If <math>x = \{0\}</math>, then the discriminant is nonzero, therefore resulting in two solutions for <math>y</math>.
 +
 
 +
Thus, the answer is <math>2 \cdot 1 + 1 \cdot 2 = \boxed{\textbf{(D) }4}</math>.
 +
 
 +
~Tiblis
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=== Solution 3: Solve for x first ===
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Set it up as a quadratic in terms of y:
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<cmath>y^2-2y+x^{2020}=0</cmath>
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Then the discriminant is
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<cmath>\Delta = 4-4x^{2020}</cmath>
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This will clearly only yield real solutions when <math>|x^{2020}| \leq 1</math>, because the discriminant must be positive.
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Then <math>x=-1,0,1</math>. Checking each one:
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<math>-1</math> and <math>1</math> are the same when raised to the 2020th power:
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<cmath>y^2-2y+1=(y-1)^2=0</cmath>
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This has only has solutions <math>1</math>, so <math>(\pm 1,1)</math> are solutions.
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Next, if <math>x=0</math>:
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<cmath>y^2-2y=0 \Rightarrow y(y-2)=0</cmath>
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Which has 2 solutions, so <math>(0,2)</math> and <math>(0,0)</math>.
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These are the only 4 solutions, so our answer is <math>\boxed{\textbf{(D) } 4}</math>.
 +
 
 +
~edits by BakedPotato66
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=== Solution 4: Solve for y first ===
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Move the <math>y^2</math> term to the other side to get <math>x^{2020}=2y-y^2 = y(2-y)</math>.
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Because <math>x^{2020} \geq 0</math> for all <math>x</math>, then <math>y(2-y) \geq 0 \Rightarrow y = 0,1,2</math>.
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If <math>y=0</math> or <math>y=2</math>, the right side is <math>0</math> and therefore <math>x=0</math>.
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When <math>y=1</math>, the right side become <math>1</math>, therefore <math>x=1,-1</math>.
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Our solutions are <math>(0,2)</math>, <math>(0,0)</math>, <math>(1,1)</math>, <math>(-1,1)</math>. These are the only <math>4</math> solutions, so the answer is <math>\boxed{\textbf{(D) } 4}</math>
 +
 
 +
- wwt7535
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 +
~ edits by BakedPotato66
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=== Solution 5: Similar to solution 4===
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Since <math>x^{2020}</math> and <math>y^2</math> are perfect squares, they are both nonnegative. That means <math>y^2</math> plus a nonnegative number equals <math>2y</math>, which means <math>y^2 \leq 2y.</math> The only possible integer values for <math>y</math> are <math>0, 1, 2</math>.
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For <math>y=0</math>, <math>x</math> can only be <math>0</math>.
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For <math>y=1</math>, <math>x^2=1</math> so <math>x=1, -1</math>.
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For <math>y=2</math>, <math>x</math> can only be <math>0</math> as well.
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 +
This gives us the solutions <math>(0, 0)</math>, <math>(1, 1)</math>, <math>(-1, 1)</math>, and <math>(0, 2)</math>. These are the only solutions, so there is a total of <math>\boxed{\textbf{(D) } 4}</math> ordered pairs.
 +
 
 +
- kc1374
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 +
=== Solution 6: (Casework)===
 +
We see that <math>x</math> has to be <math>1</math>, <math>0</math>, or <math>-1</math>, as any other integer would make this value too large. We also know that because <math>2020</math> is even, both <math>-1</math>, and <math>1</math> for <math>x</math> will yield the same <math>x^{2020}</math> value of <math>1</math>.
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'''Case 1''': <math>x=0</math>. This gives us that <math>y^2 = 2y</math>. Dividing both sides by <math>y</math> gives us <math>y = 2</math>. Additionally, we know intuitively that <math>y = 0</math> is also a case, which gives us 2 possibilities for this case.
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'''Case 2''': <math>x = 1</math> or <math>-1</math>. This gives us that <math>1 + y^2 = 2y</math>. Bringing the <math>2y</math> to the other side, we have a simple quadratic.  <math>y^2-2y + 1 = 0</math>. Factor to get <math>(y-1)^2 = 0</math> so <math>y = 1</math>. Because this works for <math>x</math> as <math>-1</math> and <math>1</math>, there are 2 possibilities for this case.
 +
 
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Adding the cases gets us our final answer of <math>\boxed{\textbf{(D) } 4}</math> ordered pairs.
 +
~iluvme
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 +
 
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=== Solution 7: (Casework)===
 +
We can move the <math>y^2</math> term to the other side to obtain <math>x^{2020} = 2y - y^2</math>, which can be easily factored into <math>x^{2020} = (2-y)y</math>. If we raise each side to the power 1/2020, we get <math>x = ((2-y)(y))^{1/2020}</math>. Since we're dealing with integer values, the inside of the square root and thus <math>(2-y)y</math> must be greater than <math>0</math>. For this to happen, <math>y</math> must be between <math>0</math> and <math>2</math> inclusive. Testing each integer value, we get the cases:
 +
 
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<math>\textbf{Case 1}</math>: When <math>y=0</math>, then <math>y(2-y)</math> is also 0, and since the square root of <math>0</math> is <math>0</math>, <math>x</math> must also be <math>0</math>. So our first pair of solutions is <math>(0,0)</math>.
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<math>\textbf{Case 2}</math>: When <math>y=1</math>, then <math>y(2-y)</math> is also 1. The possible values for <math>x</math> are <math>1</math> and <math>-1</math>, so we get 2 more pairs of solutions: <math>(1, -1)</math> and <math>(1,1)</math>.
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<math>\textbf{Case 3}</math>: When <math>y=2</math>, then <math>y(2-y)</math> is <math>0</math>, and since the square root of <math>0</math> is <math>0</math>, <math>x</math> must also be <math>0</math> - so we get one more pair of solutions, <math>(2,0)</math>
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If we count all these pairs(<math>(0,0),(1,-1),(1,1),(2,0)</math>), we have 4 pairs of solutions, so our answer is  <math>\boxed{\textbf{(D) } 4}</math> ordered pairs.
 +
 
 +
- youtube.com/indianmathguy
 +
 
 +
===Video Solution (HOW TO CREATIVELY PROBLEM SOLVE!!!)===
 +
https://youtu.be/FfATkdxncG4
 +
 
 +
~Education, the Study of Everything
 +
 
 +
===Video Solution 1 by TheBeautyOfMath===
 +
https://youtu.be/6ujfjGLzVoE
 +
 
 +
===Video Solution by WhyMath===
 +
https://youtu.be/7dQ423hhgac
 +
 
 +
~savannahsolver
  
−
~IceMatrix
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===Video Solution by Sohil Rathi===
 +
https://youtu.be/zfChnbMGLVQ?t=4251
  
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==See Also==
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== See Also ==
  
 
{{AMC10 box|year=2020|ab=B|num-b=8|num-a=10}}
 
{{AMC10 box|year=2020|ab=B|num-b=8|num-a=10}}
 +
{{AMC12 box|year=2020|ab=B|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 14:02, 16 July 2024

The following problem is from both the 2020 AMC 10B #9 and 2020 AMC 12B #8, so both problems redirect to this page.

Problem

How many ordered pairs of integers $(x, y)$ satisfy the equation \[x^{2020}+y^2=2y?\]

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

Solutions

Solution 1

Rearranging the terms and and completing the square for $y$ yields the result $x^{2020}+(y-1)^2=1$. Then, notice that $x$ can only be $0$, $1$ and $-1$ because any value of $x^{2020}$ that is greater than 1 will cause the term $(y-1)^2$ to be less than $0$, which is impossible as $y$ must be real. Therefore, plugging in the above values for $x$ gives the ordered pairs $(0,0)$, $(1,1)$, $(-1,1)$, and $(0,2)$ gives a total of $\boxed{\textbf{(D) }4}$ ordered pairs.

Solution 2

Bringing all of the terms to the LHS, we see a quadratic equation \[y^2 - 2y + x^{2020} = 0\] in terms of $y$. Applying the quadratic formula, we get \[y = \frac{2\pm\sqrt{4-4\cdot1\cdot x^{2020}}}{2}=\frac{2\pm\sqrt{4(1-x^{2020})}}{2}.\] In order for $y$ to be real, which it must be given the stipulation that we are seeking integral answers, we know that the discriminant, $4(1-x^{2020})$ must be nonnegative. Therefore, \[4(1-x^{2020}) \geq 0 \implies x^{2020} \leq 1.\] Here, we see that we must split the inequality into a compound, resulting in $-1 \leq x \leq 1$.

The only integers that satisfy this are $x \in \{-1,0,1\}$. Plugging these values back into the quadratic equation, we see that $x = \{-1,1\}$ both produce a discriminant of $0$, meaning that there is only 1 solution for $y$. If $x = \{0\}$, then the discriminant is nonzero, therefore resulting in two solutions for $y$.

Thus, the answer is $2 \cdot 1 + 1 \cdot 2 = \boxed{\textbf{(D) }4}$.

~Tiblis

Solution 3: Solve for x first

Set it up as a quadratic in terms of y: \[y^2-2y+x^{2020}=0\] Then the discriminant is \[\Delta = 4-4x^{2020}\] This will clearly only yield real solutions when $|x^{2020}| \leq 1$, because the discriminant must be positive. Then $x=-1,0,1$. Checking each one: $-1$ and $1$ are the same when raised to the 2020th power: \[y^2-2y+1=(y-1)^2=0\] This has only has solutions $1$, so $(\pm 1,1)$ are solutions. Next, if $x=0$: \[y^2-2y=0 \Rightarrow y(y-2)=0\] Which has 2 solutions, so $(0,2)$ and $(0,0)$.

These are the only 4 solutions, so our answer is $\boxed{\textbf{(D) } 4}$.

~edits by BakedPotato66

Solution 4: Solve for y first

Move the $y^2$ term to the other side to get $x^{2020}=2y-y^2 = y(2-y)$.

Because $x^{2020} \geq 0$ for all $x$, then $y(2-y) \geq 0 \Rightarrow y = 0,1,2$.

If $y=0$ or $y=2$, the right side is $0$ and therefore $x=0$.

When $y=1$, the right side become $1$, therefore $x=1,-1$.

Our solutions are $(0,2)$, $(0,0)$, $(1,1)$, $(-1,1)$. These are the only $4$ solutions, so the answer is $\boxed{\textbf{(D) } 4}$

- wwt7535

~ edits by BakedPotato66

Solution 5: Similar to solution 4

Since $x^{2020}$ and $y^2$ are perfect squares, they are both nonnegative. That means $y^2$ plus a nonnegative number equals $2y$, which means $y^2 \leq 2y.$ The only possible integer values for $y$ are $0, 1, 2$.

For $y=0$, $x$ can only be $0$.

For $y=1$, $x^2=1$ so $x=1, -1$.

For $y=2$, $x$ can only be $0$ as well.

This gives us the solutions $(0, 0)$, $(1, 1)$, $(-1, 1)$, and $(0, 2)$. These are the only solutions, so there is a total of $\boxed{\textbf{(D) } 4}$ ordered pairs.

- kc1374

Solution 6: (Casework)

We see that $x$ has to be $1$, $0$, or $-1$, as any other integer would make this value too large. We also know that because $2020$ is even, both $-1$, and $1$ for $x$ will yield the same $x^{2020}$ value of $1$.

Case 1: $x=0$. This gives us that $y^2 = 2y$. Dividing both sides by $y$ gives us $y = 2$. Additionally, we know intuitively that $y = 0$ is also a case, which gives us 2 possibilities for this case.

Case 2: $x = 1$ or $-1$. This gives us that $1 + y^2 = 2y$. Bringing the $2y$ to the other side, we have a simple quadratic. $y^2-2y + 1 = 0$. Factor to get $(y-1)^2 = 0$ so $y = 1$. Because this works for $x$ as $-1$ and $1$, there are 2 possibilities for this case.

Adding the cases gets us our final answer of $\boxed{\textbf{(D) } 4}$ ordered pairs. ~iluvme


Solution 7: (Casework)

We can move the $y^2$ term to the other side to obtain $x^{2020} = 2y - y^2$, which can be easily factored into $x^{2020} = (2-y)y$. If we raise each side to the power 1/2020, we get $x = ((2-y)(y))^{1/2020}$. Since we're dealing with integer values, the inside of the square root and thus $(2-y)y$ must be greater than $0$. For this to happen, $y$ must be between $0$ and $2$ inclusive. Testing each integer value, we get the cases:

$\textbf{Case 1}$: When $y=0$, then $y(2-y)$ is also 0, and since the square root of $0$ is $0$, $x$ must also be $0$. So our first pair of solutions is $(0,0)$.

$\textbf{Case 2}$: When $y=1$, then $y(2-y)$ is also 1. The possible values for $x$ are $1$ and $-1$, so we get 2 more pairs of solutions: $(1, -1)$ and $(1,1)$.

$\textbf{Case 3}$: When $y=2$, then $y(2-y)$ is $0$, and since the square root of $0$ is $0$, $x$ must also be $0$ - so we get one more pair of solutions, $(2,0)$

If we count all these pairs($(0,0),(1,-1),(1,1),(2,0)$), we have 4 pairs of solutions, so our answer is $\boxed{\textbf{(D) } 4}$ ordered pairs.

- youtube.com/indianmathguy

Video Solution (HOW TO CREATIVELY PROBLEM SOLVE!!!)

https://youtu.be/FfATkdxncG4

~Education, the Study of Everything

Video Solution 1 by TheBeautyOfMath

https://youtu.be/6ujfjGLzVoE

Video Solution by WhyMath

https://youtu.be/7dQ423hhgac

~savannahsolver

Video Solution by Sohil Rathi

https://youtu.be/zfChnbMGLVQ?t=4251

See Also

2020 AMC 10B (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 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 12 Problems and Solutions

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