, 2025

Let the irrational number $\alpha={1-\frac1{2a_1-\frac1{2a_2-\frac1{2a_3-\frac1{\dots}}}}}$, where coefficients $a_1$, $a_2$, $\ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha \rfloor, \lfloor 2\alpha \rfloor,\ldots , \lfloor N\alpha \rfloor$ are even. $\lfloor x \rfloor$ denotes the floor function of $x$, which is the greatest integer less than or equal to $x$.