Question 131
The numbers $S_1 = 2^2$, $S_2 = 2^4$, $\dots$, $S_n = 2^{2n}$ are given. A rectangle $OABC$ is constructed on the Cartesian plane according to these numbers. For this, starting from the point $O$, the points $A_1, A_2, \dots, A_n$ are consecutively marked along the axis $Ox$, and the points $C_1, C_2, \dots, C_n$ are consecutively marked along the axis $Oy$ in such a way that for all $k$ from $1$ to $n$ the lengths of the segments $A_{k-1}A_k = x_k$ and $C_{k-1}C_k = y_k$ are positive integers (let $A_0 = C_0 = O$, $A_n = A$ and $C_n = C$) and $x_k \cdot y_k = S_k$.
a) Find the maximal possible value of the area of the rectangle $OABC$ and all pairs of sets $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ at which this maximal area is reached.
b) Find the minimal possible value of the area of the rectangle $OABC$ and all pairs of sets $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ at which this minimal area is reached.
a) Find the maximal possible value of the area of the rectangle $OABC$ and all pairs of sets $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ at which this maximal area is reached.
b) Find the minimal possible value of the area of the rectangle $OABC$ and all pairs of sets $(x_1, x_2, \dots, x_n)$ and $(y_1, y_2, \dots, y_n)$ at which this minimal area is reached.