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 2 of 6: The region bounds x_n
Previous stepNext step
zn=⌊nβ+xn(α−β)⌋,n−20252<xn<n+20252z_n=\lfloor n\beta+x_n(\alpha-\beta)\rfloor,\qquad \frac{n-2025}{2}<x_n<\frac{n+2025}{2}
Detailed analysis

Write znz_n for the integer Emerald records at minute nn; using yn=n−xny_n=n-x_n from Step 1, zn=⌊xnα+ynβ⌋=⌊nβ+xn(α−β)⌋z_n=\lfloor x_n\alpha+y_n\beta\rfloor=\lfloor n\beta+x_n(\alpha-\beta)\rfloor. The constraint ∣xn−yn∣<2025|x_n-y_n|<2025 becomes ∣2xn−n∣<2025|2x_n-n|<2025, that is, n−20252<xn<n+20252\frac{n-2025}{2}<x_n<\frac{n+2025}{2}.