IMO 1967 LL BUL1

Prove that all numbers in the sequence

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IMO 1967 LL BUL1

Origin: BUL

Problem

Prove that all numbers in the sequence 107811 , 110778111 , 111077781111 , . . . are perfect cubes.

Solution

Let us denote the nth term of the given sequence by an. Then an = 1 103n+3 −102n+3

  • 7102n+2 −10n+1
  • 10n+2 −1  = 1 27(103n+3 −3 \cdot 102n+2 + 3 \cdot 10n+1 −1) = 10n+1 −1 3 .