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be the largest real solution to the equation ![\[\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4.\]](http://latex.artofproblemsolving.com/e/8/7/e87fdc9fe63d29a46e57dfb92c2d0448441b4b55.png)
There are positive integers 
such that 
. Find 
.
up to 
squares adjacent to it is colored which is absurd
.
coloured squares in a 
board.
coloured squares. There cannot be coloured squares in a row. Also there will be a row which contains coloured squares. squares in the 
row, then the colouring should be one of 
and 
. WLOG the second row is . We can see that there cannot be coloured squares in the 
columns. Then the others should be neighbor with the coloured ones in the row. By pigeonhole, there will be coloured squares in a column which results in a contradiction. squares in the 
or 
row, WLOG say that there are coloured squares in the first row. WLOG let the colouring be . There cannot be coloured squares in the 
columns and the second row. Also there cannot be coloured squares in the columns 
and the third row.
cannot be coloured because there's one square between and 
.
cannot be coloured because there's one square above and right or left to .
square in the last row which gives us a contradiction. coloured squares in a 
board. So there cannot be more than coloured squares in a 
board.
satisfy that: 
.
board that can be colored so that for each square in the board, at most one square adjacent to it is colored.
