Difference between revisions of "2022 AMC 8 Problems/Problem 20"
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~MrThinker | ~MrThinker | ||
+ | |||
+ | ==Solution 5 (Super fast! No algebra; no testing 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 | ||
+ | |||
+ | Please like and subscribe! | ||
==Video Solution== | ==Video Solution== | ||
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~Mathematical Dexterity | ~Mathematical Dexterity | ||
− | |||
− | |||
==Video Solution== | ==Video Solution== | ||
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~savannahsolver | ~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 22:35, 29 December 2023
Contents
- 1 Problem
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 4 (Answer Choices)
- 6 Solution 5 (Super fast! No algebra; no testing answer choices)
- 7 Video Solution by Math-X (First understand the problem!!!)
- 8 Video Solution (🚀Super Fast. Just 1 min!🚀)
- 9 Video Solution
- 10 Video Solution
- 11 Video Solution
- 12 Video Solution
- 13 Video Solution
- 14 Video Solution
- 15 See Also
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 in the lower left corner is larger than the other three missing numbers. What is the smallest possible value of
?
Solution 1
The sum of the numbers in each row is . Consider the second row. In order for the sum of the numbers in this row to equal
, the two shaded numbers must add up to
:
If two numbers add up to
, one of them must be at least
: If both shaded numbers are no more than
, their sum can be at most
. Therefore, for
to be larger than the three missing numbers,
must be at least
. We can construct a working scenario where
:
So, our answer is
.
~ihatemath123
Solution 2
The sum of the numbers in each row is and the sum of the numbers in each column is
Let be the number in the lower middle. It follows that
or
We express the other two missing numbers in terms of and
as shown below:
We have
and
Note that the first inequality is true for all values of
We only need to solve the second inequality so that the third inequality is true for all values of
By substitution, we get
from which
Therefore, the smallest possible value of is
~MRENTHUSIASM
Solution 3
This is based on the Solution 2 above and it is perhaps a little simpler than that.
Let be the number in the lower middle. Applying summation to first two columns yields the following.
Since is greater than the other three, we have
or
Therefore, the smallest possible value of is
~vetaltekdi6
Solution 4 (Answer Choices)
Note that the sum of the rows and columns must be . We proceed to test the answer choices.
Testing , when
, the number above
must be
, which contradicts the precondition that the numbers surrounding
is less than
.
Testing , the number above
is
, which does not work.
Testing , the number above
is
, which does not work.
Testing , the number above
is
, which does work. Hence, the answer is
.
We do not need to test , because the problem asks for the smallest value of
.
~MrThinker
Solution 5 (Super fast! No algebra; no testing answer choices)
The sum of the numbers in each column and row should be . If we look at the
column, the gray squares (shown below) sum to
.
If square has to be greater than or equal to the three blank squares, then the least
can be is half the sum of the value of the gray squares, which is
. But square
has to be greater than and not greater than or equal to the three blank squares, so the least
can be is
. Testing for the other rows and columns (it might be smaller than the other two squares!), we find that the smallest
can be is indeed
; the other two squares are less than
. Therefore, the answer is
~ 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!🚀)
~Education, the Study of Everything
Video Solution
Please like and subscribe!
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
~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.