Difference between revisions of "2016 AMC 10B Problems/Problem 14"
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The vertical line is just to the right of <math>x = 5</math>, the horizontal line is just under <math>y = 0</math>, and the sloped line will always be above the <math>y</math> value of <math>3x</math>. | The vertical line is just to the right of <math>x = 5</math>, the horizontal line is just under <math>y = 0</math>, and the sloped line will always be above the <math>y</math> value of <math>3x</math>. | ||
− | This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of | + | This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of <math>1\cdot1</math>, <math>2\cdot2</math>, and <math>3\cdot3</math> squares and getting <math>30</math>, <math>15</math>, and <math>5</math> respectively, and we end up with <math>\boxed{\textbf{(D)}\ 50}</math>. |
Solution by Wwang | Solution by Wwang |
Revision as of 18:03, 30 November 2019
Contents
[hide]Problem
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line , the line and the line
Solution 1
The region is a right triangle which contains the following lattice points:
Squares : Suppose that the top-right corner is , with . Then to include all other corners, we need . This produces squares.
Squares : Here . To include all other corners, we need . This produces squares.
Squares : Similarly this produces squares.
No other squares will fit in the region. Therefore the answer is .
Solution 2
The vertical line is just to the right of , the horizontal line is just under , and the sloped line will always be above the value of . This means they will always miss being on a coordinate with integer coordinates so you just have to count the number of squares to the left, above, and under these lines. After counting the number of , , and squares and getting , , and respectively, and we end up with .
Solution by Wwang
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |
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