Find all pairs of consecutive even positive integers, both of which are larger than 5,

Let x be the smaller even integer. Then, the other even integer shall be x + 2.
Therefore, as per the given condition:

$x>5$ and $x+x+2<23$

$\Rightarrow x>5$ and $2 x+2<23$

$\Rightarrow x>5$ and $2 x<21$

$\Rightarrow x>5$ and $x<\frac{21}{2}$

$\therefore x \in\{6,8,10\}$

Hence, the pairs are $(6,8),(8,10),(10,12)$.