Difference between revisions of "2023 SSMO Relay Round 3 Problems"
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Let <math>T=</math> TNYWR. Let <math>n = N+1.</math> A spray painter has a paint gun that paints all areas within a radius of <math>2.</math> The spray painter walks in the following locations, where red lines indicate red paint coming out of the gun and blue lines indicate blue paint coming out of the gun. The spray painter starts from the outermost square and works his way inwards, where in the end. The positive difference between the area of the blue-painted region and the area of the red-painted region is <math>a+b\pi.</math> Find <math>a+b.</math> (Note: if a spray painter paints an area with multiple colors, only the last color will be showing). | Let <math>T=</math> TNYWR. Let <math>n = N+1.</math> A spray painter has a paint gun that paints all areas within a radius of <math>2.</math> The spray painter walks in the following locations, where red lines indicate red paint coming out of the gun and blue lines indicate blue paint coming out of the gun. The spray painter starts from the outermost square and works his way inwards, where in the end. The positive difference between the area of the blue-painted region and the area of the red-painted region is <math>a+b\pi.</math> Find <math>a+b.</math> (Note: if a spray painter paints an area with multiple colors, only the last color will be showing). | ||
<center><asy> | <center><asy> | ||
| − | unitsize( | + | unitsize(7mm); |
label("(1,1)",(1,1.5)); | label("(1,1)",(1,1.5)); | ||
Latest revision as of 20:35, 15 December 2023
Problem 1
In triangle 
with 
, circles 
and 
are drawn, centered at 
and 
, respectively. Each of the three circles are externally tangent to the two other circles. If the radius of a circle 
such that is internally tangent to and is 
for relatively prime positive integers 
and 
, find 
Problem 2
Let 
TNYWR. In triangle with circumradius and inradius having lengths 
and 
respectively. Given that
![\[\sin\angle{A}+\sin\angle{B}+\sin\angle{C}=\left\{\sqrt{N}\right\}\]](http://latex.artofproblemsolving.com/b/9/f/b9f3c666b1307672c54c1ae2d17b8551b7e1f454.png)
the maximum value of
![\[8\sin\angle{A}\sin\angle{B}\sin\angle{C}\]](http://latex.artofproblemsolving.com/a/b/1/ab12aefc00853046c4b1a4e4807e579955d505e9.png)
is 
for squarefree 
find 
(Note that 
)
Problem 3
Let TNYWR. Let 
A spray painter has a paint gun that paints all areas within a radius of 
The spray painter walks in the following locations, where red lines indicate red paint coming out of the gun and blue lines indicate blue paint coming out of the gun. The spray painter starts from the outermost square and works his way inwards, where in the end. The positive difference between the area of the blue-painted region and the area of the red-painted region is 
Find 
(Note: if a spray painter paints an area with multiple colors, only the last color will be showing).
![[asy] unitsize(7mm); label("(1,1)",(1,1.5)); label("(-1,1)",(-1,1.5)); label("(-1,-1)",(-1,-1.5)); label("(1,-1)",(1,-1.5)); label("(2,2)",(2,2.5)); label("(-2,2)",(-2,2.5)); label("(-2,-2)",(-2,-2.5)); label("(2,-2)",(2,-2.5)); label("(3,3)",(3,3.5)); label("(-3,3)",(-3,3.5)); label("(-3,-3)",(-3,-3.5)); label("(3,-3)",(3,-3.5)); label("(N,N)",(9,9.5)); label("(-N,-N)",(-9,-9.5)); label("(-N,N)",(-9,9.5)); label("(N,-N)",(9,-9.5)); label("(N-1,N-1)",(7.3,8.5)); label("(-N+1,-N+1)",(-7.3,-8.5)); label("(-N+1,N-1)",(-7.3,8.5)); label("(N-1,-N+1)",(7.3,-8.5)); label("(N-2,N-2)",(5.5,7.5)); label("(-N+2,N+2)",(-5.5,-7.5)); label("(-N+2,N-2)",(-5.5,7.5)); label("(N-2,-N+2)",(5.5,-7.5)); draw((1,1)--(-1,1)--(-1,-1)--(1,-1)--cycle,red); draw((2,2)--(-2,2)--(-2,-2)--(2,-2)--cycle,blue); draw((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle,red); draw((7,7)--(-7,7)--(-7,-7)--(7,-7)--cycle,red); draw((8,8)--(-8,8)--(-8,-8)--(8,-8)--cycle,blue); draw((9,9)--(-9,9)--(-9,-9)--(9,-9)--cycle,red); dot((1,1),red); dot((-1,1),red); dot((1,-1),red); dot((-1,-1),red); dot((2,2),blue); dot((-2,2),blue); dot((2,-2),blue); dot((-2,-2),blue); dot((3,3),red); dot((-3,3),red); dot((3,-3),red); dot((-3,-3),red); dot((7,7),red); dot((-7,7),red); dot((7,-7),red); dot((-7,-7),red); dot((8,8),blue); dot((-8,8),blue); dot((8,-8),blue); dot((-8,-8),blue); dot((9,9),red); dot((-9,9),red); dot((9,-9),red); dot((-9,-9),red); dot((0,4)); dot((0,-4)); dot((4,0)); dot((-4,0)); dot((0,5)); dot((0,-5)); dot((5,0)); dot((-5,0)); dot((0,6)); dot((0,-6)); dot((6,0)); dot((-6,0)); [/asy]](http://latex.artofproblemsolving.com/b/a/c/bacde787969eb3c7d271531c03b201f1a9a279d2.png)
