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Difference between revisions of "2023 SSMO Relay Round 3 Problems"

(Created page with "==Problem 1== In triangle <math>ABC</math> with <math>AB=13,AC=14,BC=15</math>, circles <math>\omega_1,\omega_2,</math> and <math>\omega_3</math> are drawn, centered at <math...")
 
 
(2 intermediate revisions by the same user not shown)
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[[2023 SSMO Relay Round 1 Problems/Problem 1|Solution]]
 
[[2023 SSMO Relay Round 1 Problems/Problem 1|Solution]]
 +
 
==Problem 2==
 
==Problem 2==
  
 
Let <math>T=</math> TNYWR. In triangle <math>ABC</math> with circumradius and inradius having lengths <math>R</math> and <math>r,</math> respectively. Given that  
 
Let <math>T=</math> TNYWR. In triangle <math>ABC</math> with circumradius and inradius having lengths <math>R</math> and <math>r,</math> respectively. Given that  
−
<cmath>\sin\angle{A}+\sin\angle{B}+\sin\angle{C}=\left\{\sqrt{N}\right\}</cmath>
+
<cmath>\sin\angle{A}+\sin\angle{B}+\sin\angle{C}=\left\{\sqrt{T}\right\}</cmath>
 
the maximum value of
 
the maximum value of
 
<cmath>8\sin\angle{A}\sin\angle{B}\sin\angle{C}</cmath>
 
<cmath>8\sin\angle{A}\sin\angle{B}\sin\angle{C}</cmath>
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[[2023 SSMO Relay Round 1 Problems/Problem 2|Solution]]
 
[[2023 SSMO Relay Round 1 Problems/Problem 2|Solution]]
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==Problem 3==
 
==Problem 3==
  
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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 = T+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).
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<center><asy>
+
 
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unitsize(5mm);
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<asy>
 +
unitsize(7mm);
  
 
label("(1,1)",(1,1.5));
 
label("(1,1)",(1,1.5));
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dot((-6,0));
 
dot((-6,0));
 
</asy>
 
</asy>
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</center>
 
  
 
[[2023 SSMO Relay Round 1 Problems/Problem 3|Solution]]
 
[[2023 SSMO Relay Round 1 Problems/Problem 3|Solution]]

Latest revision as of 15:33, 15 September 2025

Problem 1

In triangle $ABC$ with $AB=13,AC=14,BC=15$, circles $\omega_1,\omega_2,$ and $\omega_3$ are drawn, centered at $A,B$ and $,C$, respectively. Each of the three circles are externally tangent to the two other circles. If the radius of a circle $\omega$ such that $\omega$ is internally tangent to $\omega_1,\omega_2,$ and $\omega_3$ is $\frac{m}{n},$ for relatively prime positive integers $m$ and $n$, find $m+n.$

Solution

Problem 2

Let $T=$ TNYWR. In triangle $ABC$ with circumradius and inradius having lengths $R$ and $r,$ respectively. Given that \[\sin\angle{A}+\sin\angle{B}+\sin\angle{C}=\left\{\sqrt{T}\right\}\] the maximum value of \[8\sin\angle{A}\sin\angle{B}\sin\angle{C}\] is $b+c\sqrt{a},$ for squarefree $a,$ find $|a+b+c|.$ (Note that $\left\{x\right\} = x - \lfloor x \rfloor$)

Solution

Problem 3

Let $T=$ TNYWR. Let $n = T+1.$ A spray painter has a paint gun that paints all areas within a radius of $2.$ 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 $a+b\pi.$ Find $a+b.$ (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]

Solution

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