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Problem 2

Let α\alpha and β\beta be positive real numbers. Emerald makes a trip in the coordinate plane, starting from the origin (0,0)(0,0). Each minute she moves one unit up or one unit to the right, restricting herself to the region ∣x−y∣<2025|x-y|<2025 in the coordinate plane. By the time she visits a point (x,y)(x,y), she writes down the integer ⌊xα+yβ⌋\lfloor x\alpha+y\beta\rfloor on it. It turns out that Emerald wrote each non-negative integer exactly once. Find all possible pairs (α,β)(\alpha,\beta) for which such a trip is possible.
Step 5 of 6: Every pair with α+β=2 is achievable
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xn=⌈n2⌉, yn=⌊n2⌋  ⟹  zn=nx_n=\left\lceil\frac{n}{2}\right\rceil,\ y_n=\left\lfloor\frac{n}{2}\right\rfloor\implies z_n=n
Detailed analysis

Conversely, assume α+β=2\alpha+\beta=2 with α≥β\alpha\ge\beta, so 0<β≤1≤α<20<\beta\le1\le\alpha<2. Take the path with xn=⌈n/2⌉x_n=\lceil n/2\rceil, yn=⌊n/2⌋y_n=\lfloor n/2\rfloor, which stays in ∣xn−yn∣≤1<2025|x_n-y_n|\le1<2025. If nn is even, zn=⌊n2(α+β)⌋=nz_n=\lfloor\frac n2(\alpha+\beta)\rfloor=n. If nn is odd, zn=⌊n+12α+n−12β⌋=⌊n+α−β2⌋=nz_n=\left\lfloor\frac{n+1}2\alpha+\frac{n-1}2\beta\right\rfloor=\left\lfloor n+\frac{\alpha-\beta}2\right\rfloor=n, since 0≤α−β<20\le\alpha-\beta<2 keeps the fractional part in [0,1)[0,1). So this path realizes every non-negative integer exactly once.