Problem

Source: 2023 Czech-Polish-Slovak Match Junior, individual p5 CPSJ

Tags: algebra



Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number $1$: after the first move, he receives the result 2, after the second move, the result is $\frac{5}{2}$, after the third move $\frac{29}{10}$, etc. After $300$ moves, Bartek receives the result $x$. Determine the largest integer not greater than $x$.