2022 AMC 10B Problems/Problem 22
- The following problem is from both the 2022 AMC 10B #22 and 2022 AMC 12B #21, so both problems redirect to this page.
Contents
Problem
Let be the set of circles in the coordinate plane that are tangent to each of the three circles with equations
,
, and
. What is the sum of the areas of all circles in
?
Solution 1
The circles match up as follows: Case
is brown, Case
is blue, Case
is green, and Case 4 is red.
Let
be circle
,
be circle
, and
be circle
.
All the circles in S are internally tangent to circle
.
There are four cases with two circles belonging to each:
and
are internally tangent to
.
and
are externally tangent to
.
is externally and Circle
is internally tangent to
.
is internally and Circle
is externally tangent to
.
Consider Cases and
together. Since circles
and
have the same center, the line connecting the center of
and the center of
will pass through the tangency point of both
and
and the tangency point of
and
. This line will be the diameter of
and have length
. Therefore the radius of
in these cases is
.
Consider Cases and
together. Similarly to Cases
and
, the line connecting the center of
to the center of
will pass through the tangency points. This time, however, the diameter of
will have length
. Therefore, the radius of
in these cases is
.
The set of circles consists of
circles -
of which have radius
and
of which have radius
.
The total area of all circles in
is
.
-naman12
Solution 2
We denote by the circle that has the equation
.
We denote by
the circle that has the equation
.
We denote by
the circle that has the equation
.
We denote by a circle that is tangent to
,
and
.
We denote by
the coordinates of circle
, and
the radius of this circle.
From the graphs of circles ,
,
, we observe that if
is tangent to all of them, then
must be internally tangent to
.
We have
We do the following casework analysis in terms of the whether is externally tangent to
and
.
Case 1: is externally tangent to
and
.
We have
and
Taking , we get
. Thus,
.
We can further compute (omitted here) that there exist feasible
with this given
.
Case 2: is internally tangent to
and
is externally tangent to
.
We have
and
Taking , we get
. Thus,
.
We can further compute (omitted here) that there exist feasible
with this given
.
Case 3: is externally tangent to
and
is internally tangent to
.
We have
and
Taking , we get
. Thus,
.
We can further compute (omitted here) that there exist feasible
with this given
.
Case 4: is internally tangent to
and
is internally tangent to
.
We have
and
Taking , we get
. Thus,
.
We can further compute (omitted here) that there exist feasible
with this given
.
Because the graph is symmetric with the -axis, and for each case above, the solution of
is not 0. Hence, in each case, there are two congruent circles whose centers are symmetric through the
-axis.
Therefore, the sum of the areas of all the circles in is
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
~MrThinker (LaTeX Error)
Video Solution by OmegaLearn using Circular Tangency
~ pi_is_3.14
Video Solution
~ThePuzzlr
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Video Solution by TheBeautyofMath
With additional justification reasoning for certain statements made. Also an additional twist on a potential similar alternate problem at the end. https://youtu.be/r-jNrjKIXTU
~IceMatrix
Video Solution by The Power of Logic(#20-#21)
See Also
2022 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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 |
2022 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.