A3. A large square consists of 2015 by 2015 small squares. The squares on the two diagonals and on the four secondary diagonals are gray, the rest is white (see the figure).

How many squares are gray?
A) 12081
B) 12082
C) 12085
D) 12086
E) 12090

![](https://cdn.mathpix.com/cropped/2024_04_17_998065f2d90ec501d25cg-1.jpg?height=306&width=325&top_left_y=2257&top_left_x=1505)
A3. A) 12081 The diagonals both contain 2015 squares. Since the dimensions of the square are odd, the diagonals intersect each other but do not intersect the secondary diagonals. Therefore, there are 2 . $2015-1=4029$ squares colored gray on the diagonals. Each of the four secondary diagonals contains 2014 squares. Two secondary diagonals that do not run in the same direction intersect each other. In other words, there are four intersection points that are 'doubly colored.' Therefore, $4 \cdot 2014-4=$ 8052 squares are colored gray on the secondary diagonals. In total, there are $4029+8052=12081$

![](https://cdn.mathpix.com/cropped/2024_04_17_20b3a516711f91b2fa07g-1.jpg?height=360&width=354&top_left_y=1399&top_left_x=1522)
squares colored gray.