Question 154
Let $a$ and $b$ be positive real numbers with different product from $1$, the star operation is defined, represented by “$*$”, by the equation $$
a * b = \frac{a + b - 2ab}{1 - ab}.
$$ On a blackboard, $2015$ numbers are written in the same number as $\frac{1}{2}$. At each step, two numbers $x$ and $y$ are erased on the board and the number $x * y$ is written. This step is repeated $2014$ times until only a number is left on the board.
a. Demonstrate that the equation $$
\frac{x * y}{1 - x * y} = \frac{x}{1 - x} + \frac{y}{1 - y}
$$ is true for any real $x$ and $y$ real with $x \neq 1$, $y \neq 1$ and $xy \neq 1$.
b. If for each number $x$ that is written on the board, we calculate $\frac{x}{1 - x}$ and add all these results, we will have a certain result. Show that this result is always the same no matter how many steps have been taken so far.
c. What number will be written on the board at the end of the $2014$ steps?
a * b = \frac{a + b - 2ab}{1 - ab}.
$$ On a blackboard, $2015$ numbers are written in the same number as $\frac{1}{2}$. At each step, two numbers $x$ and $y$ are erased on the board and the number $x * y$ is written. This step is repeated $2014$ times until only a number is left on the board.
a. Demonstrate that the equation $$
\frac{x * y}{1 - x * y} = \frac{x}{1 - x} + \frac{y}{1 - y}
$$ is true for any real $x$ and $y$ real with $x \neq 1$, $y \neq 1$ and $xy \neq 1$.
b. If for each number $x$ that is written on the board, we calculate $\frac{x}{1 - x}$ and add all these results, we will have a certain result. Show that this result is always the same no matter how many steps have been taken so far.
c. What number will be written on the board at the end of the $2014$ steps?