In a grid containing 50 points by 50 points, each point has been painted either blue or red. Directly adjacent points, whether horizontally or vertically, that are of the same color are connected by segments of the same color, while points of different colors are connected by black segments. Among the points, 1510 were blue, of which 110 were on the edge and none were in the corners. Among the segments, 947 were red. How many black and how many blue segments were drawn?
There are 990 red points in the grid, because the total number of points is 2500, and 1510 of these are blue. The placement of the red points is as follows: 4 red points are in the corners, from which $4 \cdot 2=8$ lines extend; 82 are on the edges (since there are no blue points in the corners and out of 192 edge positions, 110 are blue), from which $82 \cdot 3=246$ lines extend; the remaining $990-82-4=904$ red points are inside, from which $904 \cdot 4=3616$ lines extend. If we count the lines extending from the red points, the 947 red lines are counted twice, the black lines are counted once, and the blue lines are not counted at all, because both ends of the red lines are red, one end of the black lines is red, and neither end of the blue lines is red. This gives us exactly $8+246+3616=3870$ lines, so the number of black lines is $3870-2 \cdot 947=1976$. The number of blue lines is then, since there are a total of 4900 lines in the grid, all of which are red, blue, or black, $4900-947-1976=1977$.

We can easily verify that a coloring that meets the conditions exists. Color red the rows of the 1st, 3rd, 5th, ..., 35th columns; the 37th column from the 2nd to the 49th row; the 39th column from the 2nd to the 20th row; the 41st column rows 2, 4, ..., 42; and the 50th column's first and last elements.

Miklós Arató (Budapest, József A. Gymnasium, 1st year, o. t.)