The following solution was recieved from Andrei of School 205 Bucharest. Well done and thank you Andrei.

10201 could be written (in base x) as:

10201
= 1x0 + 2x2 + 1x4
= x4 + 2x2 + 1
= (x2 + 1)2


Now, I write 10101 in a similar manner, in base y:

10101
= y4 + y2 + 1
= y4 + 2y2 - y2 + 1
= (y4 + 2y2 + 1) - y2
= (y2 + 1)2 - y2
= (y2 + 1 + y)(y2 + 1 -y)

Therefore both expressions can be factorised, so they are composite.