MathLabs

Problem 6

Let a,b,c,d be odd integers with 0<a<b<c<d0<a<b<c<d and ad=bcad=bc. Prove that if a+d=2ka+d=2^k and b+c=2mb+c=2^m, then a=1a=1.
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2ka=22m−2⟹a=12^ka=2^{2m-2}\Longrightarrow a=1
Detailed analysis

Substitution into bc=adbc=ad simplifies to 2ka=22m−22^ka=2^{2m-2}. Since aa is odd, this equality is possible only when a=1a=1.