Difference between revisions of "2021 AMC 10B Problems/Problem 8"
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In both solutions below, note that the numbers along the yellow cells are consecutive odd perfect squares. We can show this by induction. | In both solutions below, note that the numbers along the yellow cells are consecutive odd perfect squares. We can show this by induction. | ||
Revision as of 02:23, 18 April 2021
Contents
Problem
Mr. Zhou places all the integers from to into a by grid. He places in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. What is the sum of the greatest number and the least number that appear in the second row from the top?
Solution 1
By observing that the right-top corner of a square will always be a square, we know that the top right corner of the x grid is . We can subtract to get the value of the top-left corner; . We can then find the value of the bottom left and right corners similarly. From there, we can find the value of the box on the far right in the 2nd row from the top by subtracting , since the length of the side will be one box shorter. Similarly, we find the value for the box 2nd from the left and 2nd from the top, which is . We know that the least number in the 2nd row will be , and the greatest will be the number to its left, which is less than . We then sum and to get .
-Dynosol
Solution 2: Draw It Out
Drawing out the diagram, we get . Note that this should mainly be used just to check your answer.
~Taco12
Solution 3 (Illustrations)
In both solutions below, note that the numbers along the yellow cells are consecutive odd perfect squares. We can show this by induction.
Two pictorial solutions follow from here:
Solution 3.1 (Illustration of Solution 1: Considers Only 5 Squares)
In the diagram above, the red arrows indicate the progression of numbers. In the second row from the top, the greatest and the least numbers are and respectively. By observations, we proceed as follows: Therefore, the answer is
~MRENTHUSIASM
Solution 3.2 (Illustration of Solution 2: Draws All 225 Squares Out)
From the full diagram above, the answer is
~MRENTHUSIASM
Video Solution by OmegaLearn (Using Pattern Finding)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/GYpAm8v1h-U?t=412
~IceMatrix
Video Solution by Interstigation
https://youtu.be/DvpN56Ob6Zw?t=667
~Interstigation
See Also
2021 AMC 10B (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 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.