Murashkin M.V.

On each cell of a $10 \times 10$ board, there is a chip. It is allowed to choose a diagonal with an even number of chips and remove any chip from it.

What is the maximum number of chips that can be removed from the board using such operations?
Let's call a (non-)even diagonal the one on which there is a (non-)even number of chips (at the moment). After removing a chip, an even diagonal becomes odd, and an odd one becomes even. Therefore, the number of odd diagonals does not decrease. At the beginning, there are 20 odd diagonals on the board, so there will be no fewer than 20 at the end. Among them, there are no fewer than 10 parallel ones, and on them, there will be no fewer than 10 chips left.

It is possible to remove 90 chips, for example, in the following order. Remove all the chips from the left column. Now we can remove all the chips from the 2nd column, except for the top and bottom ones, then all the chips from the 3rd column... and so on.

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