Question 29
Let $M$ and $N$ be positive integers. Mr. Pisut starts walking from the point $(0, N)$ to the point $(M, 0)$ in such a way that:
* each of his steps is of 1 unit length in the direction parallel to either the X-axis or the Y-axis;
* for each point $(x, y)$ on his path, $x \ge 0$ and $y \ge 0$.
For each step, he measures the distance from himself to the axis to which his step is parallel. If the step takes him farther away from the origin, he records the distance as a positive value; otherwise it is recorded in negative.
Prove that after he finishes his walk, the sum of all distances recorded is zero.
* each of his steps is of 1 unit length in the direction parallel to either the X-axis or the Y-axis;
* for each point $(x, y)$ on his path, $x \ge 0$ and $y \ge 0$.
For each step, he measures the distance from himself to the axis to which his step is parallel. If the step takes him farther away from the origin, he records the distance as a positive value; otherwise it is recorded in negative.
Prove that after he finishes his walk, the sum of all distances recorded is zero.