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Difference between revisions of "2022 AMC 8 Problems/Problem 20"

m (→‎See Also)
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== Solution 1 ==
 
== Solution 1 ==
−
The sum of the numbers in each row is <math>12</math>. Consider the second row. In order for the sum of the numbers in this row to equal <math>12</math>, the first two numbers must add up to <math>13</math>:
+
The sum of the numbers in each row is <math>12</math>. Consider the second row. In order for the sum of the numbers in this row to equal <math>12</math>, the two shaded numbers must add up to <math>13</math>:
 
<asy>
 
<asy>
 
unitsize(0.5cm);
 
unitsize(0.5cm);
−
fill((-3,1)--(1,1)--(1,-1)--(-3,-1)--cycle,lightgray);
+
fill((-3,1)--(1,1)--(1,-1)--(-3,-1)--cycle,mediumgray);
 
draw((3,3)--(-3,3));
 
draw((3,3)--(-3,3));
 
draw((3,1)--(-3,1));
 
draw((3,1)--(-3,1));
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label((2,-2),"$8$");
 
label((2,-2),"$8$");
 
label((-2,-2),"$x$");
 
label((-2,-2),"$x$");
−
label((0,-2),"$y$",red); label((-2,0),"$y+10$",red+fontsize(8)); label((0,0),"$x-1$",red+fontsize(8));  
+
label((0,-2),"$y$",red+fontsize(11)); label((-2,0),"$y{+}10$",red+fontsize(11)); label((0,0),"$x{-}1$",red+fontsize(11));  
 
</asy>
 
</asy>
 
We have <math>x>x-1, x>y+10,</math> and <math>x>y.</math> Note that the first inequality is true for all values of <math>x.</math> We only need to solve the second inequality so that the third inequality is true for all values of <math>x.</math> By substitution, we get <math>x>(4-x)+10,</math> from which <math>x>7.</math>
 
We have <math>x>x-1, x>y+10,</math> and <math>x>y.</math> Note that the first inequality is true for all values of <math>x.</math> We only need to solve the second inequality so that the third inequality is true for all values of <math>x.</math> By substitution, we get <math>x>(4-x)+10,</math> from which <math>x>7.</math>
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~MRENTHUSIASM
 
~MRENTHUSIASM
 +
 +
==Solution 3==
 +
 +
This is based on the Solution 2 above and it is perhaps a little simpler than that.
 +
 +
Let <math>y</math> be the number in the lower middle. Applying summation to first two columns yields the following.
 +
 +
<asy>
 +
unitsize(0.5cm);
 +
draw((3,3)--(-3,3));
 +
draw((3,1)--(-3,1));
 +
draw((3,-3)--(-3,-3));
 +
draw((3,-1)--(-3,-1));
 +
draw((3,3)--(3,-3));
 +
draw((1,3)--(1,-3));
 +
draw((-3,3)--(-3,-3));
 +
draw((-1,3)--(-1,-3));
 +
label((-2,2),"$-2$");
 +
label((0,2),"$9$");
 +
label((2,2),"$5$");
 +
label((2,0),"$-1$");
 +
label((2,-2),"$8$");
 +
label((-2,-2),"$x$");
 +
label((0,-2),"$y$",red+fontsize(11)); label((-2,0),"$14{-}x$",red+fontsize(11)); label((0,0),"$3{-}y$",red+fontsize(11));
 +
</asy>
 +
 +
Since <math>x</math> is greater than the other three, we have <math>x>14-x,</math> or <math>x>7.</math>
 +
 +
Therefore, the smallest possible value of <math>x</math> is <math>\boxed{\textbf{(D) } 8}.</math>
 +
 +
~vetaltekdi6
 +
 +
==Solution 4 (Answer Choices)==
 +
Note that the sum of the rows and columns must be <math>8+5-1=12</math>. We proceed to test the answer choices.
 +
 +
Testing <math>\textbf{(A)}</math>, when <math>x = -1</math>, the number above <math>x</math> must be <math>15</math>, which contradicts the precondition that the numbers surrounding <math>x</math> is less than <math>x</math>.
 +
 +
Testing <math>\textbf{(B)}</math>, the number above <math>x</math> is <math>9</math>, which does not work.
 +
 +
Testing <math>\textbf{(C)}</math>, the number above <math>x</math> is <math>8</math>, which does not work.
 +
 +
Testing <math>\textbf{(D)}</math>, the number above <math>x</math> is <math>6</math>, which ''does'' work. Hence, the answer is <math>\boxed{\textbf{(D) }8}</math>.
 +
 +
We do not need to test <math>\textbf{(E)}</math>, because the problem asks for the '''smallest''' value of <math>x</math>.
 +
 +
~MrThinker
 +
 +
==Solution 5 (Super fast! No algebra and no testing any of the answer choices)==
 +
 +
The sum of the numbers in each column and row should be <math>5+(-1)+8=12</math>. If we look at the <math>1^{\text{st}}</math> column, the gray squares (shown below) sum to <math>12-(-2)=14</math>.
 +
 +
<asy>
 +
draw((3,3)--(-3,3));
 +
draw((3,1)--(-3,1));
 +
draw((3,-3)--(-3,-3));
 +
draw((3,-1)--(-3,-1));
 +
draw((3,3)--(3,-3));
 +
draw((1,3)--(1,-3));
 +
draw((-3,3)--(-3,-3));
 +
draw((-1,3)--(-1,-3));
 +
label((-2,2),"$-2$"); 
 +
label((0,2),"$9$");
 +
label((2,2),"$5$");
 +
label((2,0),"$-1$");
 +
label((2,-2),"$8$");
 +
label((-2,-2),"$x$");
 +
filldraw((-3,-3)--(-1,-3)--(-1,-1)--(-3,-1)--cycle, lightgray, black+linewidth(1));
 +
filldraw((-1,-1)--(-3,-1)--(-3,1)--(-1,1)--cycle, lightgray, black+linewidth(1));
 +
label(scale(1)*"All credits for original unedited asymptote for the problem go to whoever made the asymptote in the 'Problem' section.", (-0,-5), S);
 +
</asy>
 +
 +
'''If''' square <math>x</math> has to be '''greater than or equal to''' the three blank squares, then the least <math>x</math> can be is half the sum of the value of the gray squares, which is <math>14\div2=7</math>. But square <math>x</math> has to be '''greater than''' and '''''not''''' '''greater than or equal to''' the three blank squares, so the least <math>x</math> can be is <math>7+1=8</math>. Testing for the other rows and columns (it might be smaller than the other two squares!), we find that the smallest <math>x</math> can be is indeed <math>8</math>; the other two squares are less than <math>8</math>. Therefore, the answer is <math>\boxed{\text{(D) }8}</math>.
 +
 +
~ JoyfulSapling
 +
 +
==Video Solution by Math-X (First understand the problem!!!)==
 +
https://youtu.be/oUEa7AjMF2A?si=Bbea8RWE2sMWN6xl&t=3643
 +
 +
~Math-X
 +
 +
==Video Solution (🚀Super Fast. Just 1 min!🚀)==
 +
https://youtu.be/7J4EGPaB29Y
 +
 +
<i>~Education, the Study of Everything</i>
 +
 +
==Video Solution==
 +
https://youtu.be/0hHlpIVeFjg
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=xnGQffaxYAA
 +
 +
~Mathematical Dexterity
 +
 +
==Video Solution==
 +
https://youtu.be/Ij9pAy6tQSg?t=1857
 +
 +
~Interstigation
 +
 +
==Video Solution==
 +
https://youtu.be/hs6y4PWnoWg?t=369
 +
 +
~STEMbreezy
 +
 +
==Video Solution==
 +
https://youtu.be/DXFwzrOF4b4
 +
 +
~savannahsolver
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=FINk9LgSJpU
 +
 +
~David
  
 
==See Also==  
 
==See Also==  
 
{{AMC8 box|year=2022|num-b=19|num-a=21}}
 
{{AMC8 box|year=2022|num-b=19|num-a=21}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 12:10, 14 October 2024

Problem

The grid below is to be filled with integers in such a way that the sum of the numbers in each row and the sum of the numbers in each column are the same. Four numbers are missing. The number $x$ in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of $x$? [asy] unitsize(0.5cm); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); [/asy] $\textbf{(A) } -1 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 8 \qquad \textbf{(E) } 9 \qquad$

Solution 1

The sum of the numbers in each row is $12$. Consider the second row. In order for the sum of the numbers in this row to equal $12$, the two shaded numbers must add up to $13$: [asy] unitsize(0.5cm); fill((-3,1)--(1,1)--(1,-1)--(-3,-1)--cycle,mediumgray); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); [/asy] If two numbers add up to $13$, one of them must be at least $7$: If both shaded numbers are no more than $6$, their sum can be at most $12$. Therefore, for $x$ to be larger than the three missing numbers, $x$ must be at least $8$. We can construct a working scenario where $x=8$: [asy] unitsize(0.5cm); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$8$"); label((0,-2),"$-4$"); label((-2,0),"$6$"); label((0,0),"$7$"); [/asy] So, our answer is $\boxed{\textbf{(D) } 8}$.

~ihatemath123

Solution 2

The sum of the numbers in each row is $-2+9+5=12,$ and the sum of the numbers in each column is $5+(-1)+8=12.$

Let $y$ be the number in the lower middle. It follows that $x+y+8=12,$ or $x+y=4.$

We express the other two missing numbers in terms of $x$ and $y,$ as shown below: [asy] unitsize(0.5cm); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); label((0,-2),"$y$",red+fontsize(11)); label((-2,0),"$y{+}10$",red+fontsize(11)); label((0,0),"$x{-}1$",red+fontsize(11)); [/asy] We have $x>x-1, x>y+10,$ and $x>y.$ Note that the first inequality is true for all values of $x.$ We only need to solve the second inequality so that the third inequality is true for all values of $x.$ By substitution, we get $x>(4-x)+10,$ from which $x>7.$

Therefore, the smallest possible value of $x$ is $\boxed{\textbf{(D) } 8}.$

~MRENTHUSIASM

Solution 3

This is based on the Solution 2 above and it is perhaps a little simpler than that.

Let $y$ be the number in the lower middle. Applying summation to first two columns yields the following.

[asy] unitsize(0.5cm); draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); label((0,-2),"$y$",red+fontsize(11)); label((-2,0),"$14{-}x$",red+fontsize(11)); label((0,0),"$3{-}y$",red+fontsize(11)); [/asy]

Since $x$ is greater than the other three, we have $x>14-x,$ or $x>7.$

Therefore, the smallest possible value of $x$ is $\boxed{\textbf{(D) } 8}.$

~vetaltekdi6

Solution 4 (Answer Choices)

Note that the sum of the rows and columns must be $8+5-1=12$. We proceed to test the answer choices.

Testing $\textbf{(A)}$, when $x = -1$, the number above $x$ must be $15$, which contradicts the precondition that the numbers surrounding $x$ is less than $x$.

Testing $\textbf{(B)}$, the number above $x$ is $9$, which does not work.

Testing $\textbf{(C)}$, the number above $x$ is $8$, which does not work.

Testing $\textbf{(D)}$, the number above $x$ is $6$, which does work. Hence, the answer is $\boxed{\textbf{(D) }8}$.

We do not need to test $\textbf{(E)}$, because the problem asks for the smallest value of $x$.

~MrThinker

Solution 5 (Super fast! No algebra and no testing any of the answer choices)

The sum of the numbers in each column and row should be $5+(-1)+8=12$. If we look at the $1^{\text{st}}$ column, the gray squares (shown below) sum to $12-(-2)=14$.

[asy] draw((3,3)--(-3,3)); draw((3,1)--(-3,1)); draw((3,-3)--(-3,-3)); draw((3,-1)--(-3,-1)); draw((3,3)--(3,-3)); draw((1,3)--(1,-3)); draw((-3,3)--(-3,-3)); draw((-1,3)--(-1,-3)); label((-2,2),"$-2$"); label((0,2),"$9$"); label((2,2),"$5$"); label((2,0),"$-1$"); label((2,-2),"$8$"); label((-2,-2),"$x$"); filldraw((-3,-3)--(-1,-3)--(-1,-1)--(-3,-1)--cycle, lightgray, black+linewidth(1)); filldraw((-1,-1)--(-3,-1)--(-3,1)--(-1,1)--cycle, lightgray, black+linewidth(1)); label(scale(1)*"All credits for original unedited asymptote for the problem go to whoever made the asymptote in the 'Problem' section.", (-0,-5), S); [/asy]

If square $x$ has to be greater than or equal to the three blank squares, then the least $x$ can be is half the sum of the value of the gray squares, which is $14\div2=7$. But square $x$ has to be greater than and not greater than or equal to the three blank squares, so the least $x$ can be is $7+1=8$. Testing for the other rows and columns (it might be smaller than the other two squares!), we find that the smallest $x$ can be is indeed $8$; the other two squares are less than $8$. Therefore, the answer is $\boxed{\text{(D) }8}$.

~ JoyfulSapling

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=Bbea8RWE2sMWN6xl&t=3643

~Math-X

Video Solution (🚀Super Fast. Just 1 min!🚀)

https://youtu.be/7J4EGPaB29Y

~Education, the Study of Everything

Video Solution

https://youtu.be/0hHlpIVeFjg

Video Solution

https://www.youtube.com/watch?v=xnGQffaxYAA

~Mathematical Dexterity

Video Solution

https://youtu.be/Ij9pAy6tQSg?t=1857

~Interstigation

Video Solution

https://youtu.be/hs6y4PWnoWg?t=369

~STEMbreezy

Video Solution

https://youtu.be/DXFwzrOF4b4

~savannahsolver

Video Solution

https://www.youtube.com/watch?v=FINk9LgSJpU

~David

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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 AJHSME/AMC 8 Problems and Solutions

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

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