Question 95
João was studying for the Mathematical Olympiads and came across the following equation
$$
\frac{1}{x} + \frac{1}{y} = \frac{1}{2} + \frac{1}{z}
$$
where $x$, $y$ and $z$ are positive integers. After trying unsuccessfully to find all the solutions, he asked for help from Professor Piraldo, who decided to give him some hints on how to proceed. Let us help João interpret the hints.
a. If $x=1$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
b. If $x=2$ and $y \geq 2$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
c. If $x=3$ and $y \geq 3$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
d. If $x$ and $y$ are greater than or equal to 4, verify that the equation has no solution.
$$
\frac{1}{x} + \frac{1}{y} = \frac{1}{2} + \frac{1}{z}
$$
where $x$, $y$ and $z$ are positive integers. After trying unsuccessfully to find all the solutions, he asked for help from Professor Piraldo, who decided to give him some hints on how to proceed. Let us help João interpret the hints.
a. If $x=1$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
b. If $x=2$ and $y \geq 2$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
c. If $x=3$ and $y \geq 3$, determine all pairs $(y, z)$ of positive integers that satisfy the equation.
d. If $x$ and $y$ are greater than or equal to 4, verify that the equation has no solution.