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Pairwise differences form geometric sequence

Source: ISL 2022 A5

July 9, 2023

algebrageometric sequence

Problem Statement

Find all positive integers

n \geqslant 2

for which there exist

n

real numbers

a_1<\cdots<a_n

and a real number

r>0

such that the

\tfrac{1}{2}n(n-1)

differences

a_j-a_i

for

1 \leqslant i<j \leqslant n

are equal, in some order, to the numbers

r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}

.

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