Estonian Mathematical Olympiad

In a common year, there are $365=13\cdot 28+1$ days. Thus, after a common year, the dates of the airings shift 1 day earlier. After a leap year, the airings shift 2 days earlier. In every 4 years, there are 3 common years and 1 leap year, which means a total shift of $3\cdot 1+2=5$ days. Thus in $5\cdot 4=20$ years the total shift is $5\cdot 5=25$ days. Thus, if in year $a$ the show airs on January 29, then in year $a+20$ it will air on January 4, and in the intermediate years it will air between these dates. In these years, there cannot be a second airing in January. However, in year $a+21$, the first airing will be on January 2 or January 3, which also means a second airing in January. Thus 21 years is the desired answer.