MathLabs

Problem 1

Prove that for each real number r>2r>2, there are exactly two or three positive real numbers xx satisfying the equation x2=r⌊x⌋x^2=r\lfloor x\rfloor. (Here ⌊x⌋\lfloor x\rfloor denotes the largest integer less than or equal to xx.)
Step 4 of 4: Conclude
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2≤#{x}≤32\le \#\{x\}\le 3
Detailed analysis

The two integers k=⌊r⌋,⌊r⌋−1k=\lfloor r\rfloor,\lfloor r\rfloor-1 from the previous step both satisfy r−2<k≤r⊂(r−3,r]r-2<k\le r\subset(r-3,r], so among the at most three candidates found earlier, at least these two always give genuine solutions. Hence the number of positive solutions xx is at least 22 and at most 33, i.e. exactly two or three.