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 1 of 6: The coordinates always sum to the step count
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xn+yn=nx_n+y_n=n
Detailed analysis

Let (xn,yn)(x_n,y_n) be the point visited after nn minutes. Each minute increases exactly one coordinate by 11, so xn+1+yn+1=xn+yn+1x_{n+1}+y_{n+1}=x_n+y_n+1; since x0+y0=0x_0+y_0=0, this gives xn+yn=nx_n+y_n=n for every nn.