Problem 2

Let

\alpha

and

\beta

be positive real numbers. Emerald makes a trip in the coordinate plane, starting from the origin

(0,0)

. Each minute she moves one unit up or one unit to the right, restricting herself to the region

|x-y|<2025

in the coordinate plane. By the time she visits a point

(x,y)

, she writes down the integer

\lfloor x\alpha+y\beta\rfloor

on it. It turns out that Emerald wrote each non-negative integer exactly once. Find all possible pairs

(\alpha,\beta)

for which such a trip is possible.

Step 3 of 6: z_n is nondecreasing and enumerates every integer, so z_n=n

Previous step Next step

z_{n+1}\ge z_n\ \text{and}\ \{z_n\}=\mathbb{Z}_{\ge0}\implies z_n=n

Detailed analysis

Since $\alpha,\beta>0$ , moving from $(x_n,y_n)$ to $(x_{n+1},y_{n+1})$ increases $x\alpha+y\beta$ by $\alpha$ or $\beta$ , both positive, so $z_{n+1}\ge z_n$ . Since $z_0=0$ and the sequence $(z_n)$ takes every non-negative integer exactly once while being nondecreasing, it must increase by exactly one step at a time: $z_n=n$ for every $n$ .