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 6 of 6: Conclude the answer
Previous stepNext step
{(α,β):α,β>0, α+β=2}\{(\alpha,\beta):\alpha,\beta>0,\ \alpha+\beta=2\}
Detailed analysis

Step 4 shows α+β=2\alpha+\beta=2 is necessary, and Step 5 shows it is sufficient. Hence a valid trip exists exactly for the pairs (α,β)(\alpha,\beta) of positive reals with α+β=2\alpha+\beta=2.