Authors: Bogdanov I.I., Skopenkov A.B.

In space, there are 43 points: 3 yellow and 40 red. No four of them lie in the same plane. Can the number of triangles with red vertices, linked with a triangle with yellow vertices, be equal to $2023$?

A yellow triangle is linked with a red one if the contour of the red triangle intersects the part of the plane bounded by the yellow triangle exactly at one point. Triangles differing by the permutation of vertices are considered the same.
Connect each pair of red points lying on opposite sides of the yellow plane (passing through three yellow points) with a segment, and paint the segments intersecting the yellow triangle at an internal point black, and the rest white. Obviously, the number of red triangles entangled with the yellow one is equal to the number of pairs of segments of different colors having a common vertex. Let's call such a pair a "V". If the number of red points on each side of the yellow plane is odd, then the number of black and white segments emanating from each red point have different parity, so each red point is a vertex for an even number of "Vs". If the number of red points on each side of the yellow plane is even, then consider a graph,
whose vertices are the red points, and edges are the black segments. The number of vertices of odd degree in it is even, therefore, the total number of "Vs" will again be even.

## Answer

No.