Bogdanov 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 the "border" of width 5 in a $200 \times 200$ square. This border consists of four corner squares $5 \times 5$ and four rectangles $5 \times 190$. Place 3800 chips in these four rectangles: red chips in the left and top rectangles, and blue chips in the right and bottom rectangles. 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 at least one 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 the remaining nine rows contain no more than $9 \cdot 200 = 1800$ chips, totaling no more than $1910 + 1800 = 3710$ chips. Similarly, in the case of 191 multicolored columns. If, however, there are no more than 190 multicolored rows and no more than 190 multicolored columns, 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.

## Problem