Bogognov i.i.

In some cells of a $200 \times 200$ square, there is one chip - either red or blue; the other cells are empty. One chip sees another if they are in the same row or column. It is known that each chip sees exactly five chips of the other color (and possibly some chips of its own color). Find the maximum possible number of chips.
Example. Let's highlight a "border" of width 5 around a 200×200 square. This border consists of four corner squares $5 \times 5$ and four rectangles $5 \times 190$. Place 3800 chips in these four rectangles: in the left and top ones - red, and in the right and bottom ones - blue. It is easy to see that all the requirements are met.

Estimate. Consider an arbitrary arrangement of chips that satisfies the conditions. We will call a row (a row or a column) multicolored if it contains chips of both colors.

By the condition, each chip sees some chip of the other color, so each chip lies in at least one multicolored row. Moreover, since a multicolored row contains a red chip, it contains no more than five blue chips. Similarly, a multicolored row contains no more than five red chips, that is, no more than 10 chips in total.

If there are 191 multicolored rows, then they contain no more than $191 \cdot 10=1910$ chips, and in the remaining nine rows, no more than $9 \cdot 200=1800$ chips, totaling no more than $1910+1800=3710$ chips. Similarly, in the case when there are 191 multicolored columns. If, however, there are no more than 190 of each, then they contain no more than $190 \cdot 10+190 \cdot 10=$ 3800 chips, and all chips are contained in these rows.

## Answer

3800 chips.