users, some pairs of whom are friends. Whenever user 
is friends with user 
,
such that 
users have 
friends each, and 
and 
.
and natural 
such that 
and 
,![$\sqrt{ad^3 g^4} +\sqrt[3]{be^3 h^4} + \sqrt[8]{cf^3 k^4}$](http://latex.artofproblemsolving.com/b/f/a/bfa17e8bb4cc05a84ed6a757f506d6d0ed2b0081.png)
Y by
Consider 
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.
Find the number of position of cell 
Note: tetromino can be rotated but not flipped
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino.Find the number of position of
cell 
Note: tetromino can be rotated but not flippedThis post has been edited 1 time. Last edited by ItzsleepyXD, Apr 30, 2025, 9:35 AM
Reason: Rutthee look like L tetromino
Reason: Rutthee look like L tetromino
is nice, then both 
and 
are odd. To see this, color all cells in columns with odd indices black and all others white. Observe that every L-tetromino always covers an odd number of black cells. Since the number of L-tetrominoes used to tile the board is even, the total number of black cells covered must also be even. However, the total number of black cells on the board is odd. Thus, to balance parity, the cell removed must be black, that is, it must lie in an odd-numbered column. A symmetric argument applied to rows implies that the removed cell must also lie in an odd-numbered row. Hence, if 
board and that cell is one of the four corners or the center, then the remaining board can be tiled by L-tetrominoes.
sub-board at the corner of a 
board. Then the remaining region can be tiled by L-tetrominoes.
square can be tiled by L-tetrominoes. Removing such a 
square. If the removed cell lies in the center or a corner of some 
![\[
\left\lceil \frac{2025}{2} \right\rceil^2 = 1013^2 = 1026169
\]](http://latex.artofproblemsolving.com/6/5/f/65f0eb723ff31ab0cfb83c5b538fc6aa6235ada4.png)
from a 
