csat-suneung 2011 Q10

csat-suneung · South-Korea · csat__math-science 4 marks Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series)
There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, determine two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.
Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$, and determine two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, determine two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process to obtain $S _ { n }$, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? [4 points]
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$ 
There is a rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ with $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 }$ be the midpoint of segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$, and on segment $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$, determine two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ such that $\angle \mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 } = \angle \mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 } = 15 ^ { \circ } , \angle \mathrm { B } _ { 2 } \mathrm { M } _ { 1 } \mathrm { C } _ { 2 } = 60 ^ { \circ }$. Let $S _ { 1 }$ be the sum of the area of triangle $\mathrm { A } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 2 }$ and the area of triangle $\mathrm { C } _ { 2 } \mathrm { M } _ { 1 } \mathrm { D } _ { 1 }$.

Quadrilateral $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is a rectangle with $\overline { \mathrm { B } _ { 2 } \mathrm { C } _ { 2 } } = 2 \overline { \mathrm {~A} _ { 2 } \mathrm {~B} _ { 2 } }$, and determine two points $\mathrm { A } _ { 2 } , \mathrm { D } _ { 2 }$ as shown in the figure. Let $\mathrm { M } _ { 2 }$ be the midpoint of segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$, and on segment $\mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$, determine two points $\mathrm { B } _ { 3 } , \mathrm { C } _ { 3 }$ such that $\angle \mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 } = \angle \mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 } = 15 ^ { \circ }$, $\angle \mathrm { B } _ { 3 } \mathrm { M } _ { 2 } \mathrm { C } _ { 3 } = 60 ^ { \circ }$. Let $S _ { 2 }$ be the sum of the area of triangle $\mathrm { A } _ { 2 } \mathrm { M } _ { 2 } \mathrm {~B} _ { 3 }$ and the area of triangle $\mathrm { C } _ { 3 } \mathrm { M } _ { 2 } \mathrm { D } _ { 2 }$. Continuing this process to obtain $S _ { n }$, what is the value of $\sum _ { n = 1 } ^ { \infty } S _ { n }$? [4 points]\\
(1) $\frac { 2 + \sqrt { 3 } } { 6 }$\\
(2) $\frac { 3 - \sqrt { 3 } } { 2 }$\\
(3) $\frac { 4 + \sqrt { 3 } } { 9 }$\\
(4) $\frac { 5 - \sqrt { 3 } } { 5 }$\\
(5) $\frac { 7 - \sqrt { 3 } } { 8 }$
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 (Calculus) Q26 (Discrete Mathematics) Q26 (Probability and Statistics) Q27 (Calculus) Q27 (Discrete Mathematics) Q27 (Probability and Statistics) Q28 (Calculus) Q28 (Probability and Statistics) Q29 (Calculus) Q29 (Probability and Statistics) Q30 (Calculus) Q30 (Probability and Statistics)