MathDB

Let

a,b

be positive real numbers satisfying

2ab=a-b

. Denote for any positive integer

k

x_k

and

y_k

to be the closest integer to

ak

and

bk

, respectively (if there are two closest integers, choose the larger one). Prove that any positive integer

n

appears in the sequence

(x_k)_{k\ge 1}

if and only if it appears at least three times in the sequence

(y_k)_{k\ge 1}

Problem Statement