A sequence \( a_1, a_2, \ldots \) of real numbers satisfies the following condition: for every positive integer \( k \geq 2 \), there exists a positive integer \( i < k \) such that \( a_i + a_k = k \). It is known that for some \( j \), the fractional parts of the numbers \( a_j \) and \( a_{j+1} \) are equal. Prove that for some positive integers \( x \neq y \), the equality \[ a_x - a_y = x - y \]holds. The fractional part of a real number \( a \) is defined as the number \( \{a\} \in [0, 1) \), which satisfies the condition \( a = n + \{a\} \), where \( n \) is an integer. For example, \( \{-3\} = 0 \), \( \{3.14\} = 0.14 \), and \( \{-3.14\} = 0.86 \). Proposed by Mykhailo Shtandenko