MathLabs

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 4 of 6: Boundedness forces α+β=2
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n∣1−α+β2∣<20252(α−β)+1 for all n  ⟹  α+β=2n\left|1-\frac{\alpha+\beta}{2}\right|<\frac{2025}{2}(\alpha-\beta)+1\ \text{for all }n\implies\alpha+\beta=2
Detailed analysis

Assume WLOG α≥β\alpha\ge\beta. Substituting the bound on xnx_n from Step 2 into zn=⌊nβ+xn(α−β)⌋z_n=\lfloor n\beta+x_n(\alpha-\beta)\rfloor gives ∣zn−nα+β2∣<20252(α−β)+1\left|z_n-n\frac{\alpha+\beta}{2}\right|<\frac{2025}{2}(\alpha-\beta)+1 for every nn. Using zn=nz_n=n from Step 3, this reads n∣1−α+β2∣<20252(α−β)+1n\left|1-\frac{\alpha+\beta}{2}\right|<\frac{2025}{2}(\alpha-\beta)+1 for every positive integer nn, and the right side is a constant independent of nn. If α+β≠2\alpha+\beta\neq2 the left side would grow without bound as n→∞n\to\infty, a contradiction; hence α+β=2\alpha+\beta=2.