csat-suneung 2021 Q14

csat-suneung · South-Korea · csat__math-science 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series)
As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$. In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 441 } { 103 }$
(2) $\frac { 441 } { 109 }$
(3) $\frac { 441 } { 115 }$
(4) $\frac { 441 } { 121 }$
(5) $\frac { 441 } { 127 }$ 
As shown in the figure, there is a rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { AB } _ { 1 } } = 2$ and $\overline { \mathrm { AD } _ { 1 } } = 4$. Let $\mathrm { E } _ { 1 }$ be the point that divides segment $\mathrm { AD } _ { 1 }$ internally in the ratio $3 : 1$, and let $\mathrm { F } _ { 1 }$ be a point inside rectangle $\mathrm { AB } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ such that $\overline { \mathrm { F } _ { 1 } \mathrm { E } _ { 1 } } = \overline { \mathrm { F } _ { 1 } \mathrm { C } _ { 1 } }$ and $\angle \mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } = \frac { \pi } { 2 }$. Triangle $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 }$ is drawn. The figure obtained by shading quadrilateral $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ is called $R _ { 1 }$.\\
In figure $R _ { 1 }$, a rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { B } _ { 2 }$ on segment $\mathrm { AB } _ { 1 }$, point $\mathrm { C } _ { 2 }$ on segment $\mathrm { E } _ { 1 } \mathrm {~F} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on segment $\mathrm { AE } _ { 1 }$, and point A, such that $\overline { \mathrm { AB } _ { 2 } } : \overline { \mathrm { AD } _ { 2 } } = 1 : 2$. Using the same method as for obtaining figure $R _ { 1 }$, triangle $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 }$ is drawn in rectangle $\mathrm { AB } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and quadrilateral $\mathrm { E } _ { 2 } \mathrm {~F} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is shaded to obtain figure $R _ { 2 }$.\\
Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]\\
(1) $\frac { 441 } { 103 }$\\
(2) $\frac { 441 } { 109 }$\\
(3) $\frac { 441 } { 115 }$\\
(4) $\frac { 441 } { 121 }$\\
(5) $\frac { 441 } { 127 }$
Paper Questions
Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28 Q29