Can a 75x75 table (consisting of 75x75 identical square grids) be partitioned into dominoes (1x2 rectangles) and crosses (five square figures consisting of a square and its four neighbors.)
Provide adequate reasoning for your answer.
Source: St. Petersburg City Mathematical Olympiad (Russia)
Ten of the other pentominos can be chosen in place of the X pentomino in the problem, and they will all be able to tile the 75x75 table along with dominos.
The I pentomino tiles the table in a trivial manner.
A 3x3 square tiles the table, so the P, T, V, Z pemtominos can using the following configurations (A,B are dominos):
AAB ATB VAA ZZA
PPB ATB VBB BZA
PPP TTT VVV BZZ
A 3x5 rectangle tiles the table, so the F, L, N, U, Y pentominos can using the following configurations (A,B,C,D,E are dominos):
AFBBC AABCD ABBCD AABCC ABBCC
AFFFC ELBCD ANNCD DUBUE AYDDE
DDFEE ELLLL EENNN DUUUE YYYYE
This leaves the W pentomino, which I do not have an answer for yet.