2021 AMC 10B Problems/Problem 7
Contents
[hide]Problem
In a plane, four circles with radii and are tangent to line at the same point but they may be on either side of . Region consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region ?
Solution 1
After a bit of wishful thinking and inspection, we find that the above configuration maximizes our area, which is
~ samrocksnature
Solution 2 (Explains Solution 1 Using Intuition)
Suppose each circle lies north or south to line We construct the circles one by one:
1. Without the loss of generality, we draw the circle with radius north to
2. To maximize the area of the desired region, we draw the circle with radius south to
3. Now, we need to subtract out the circle with radius at least. The optimal situation is that the circle with radius encompasses the circle with radius so that we do not need to subtract more. That is, the two smallest circles are on the same side of but can be on either side. The diagram in Solution 1 shows one possible positions of the four circles.
Together, the answer is
~MRENTHUSIASM
Video Solution by OmegaLearn (Area of Circles and Logic)
~ pi_is_3.14
Video Solution by TheBeautyofMath
https://youtu.be/GYpAm8v1h-U?t=206
~IceMatrix
Video Solution by Interstigation
https://youtu.be/DvpN56Ob6Zw?t=555
~Interstigation
See Also
2021 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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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