Quiz

Damn, I was writing a response, but the message didn't send. Anyway, the main thing we need is that the points are not of a single specific color, but simply identical (otherwise, the answer is no, as we also have a case where the entire plane is green, and it is impossible to find 2 magenta points on it). So, we are looking for the boundary between two differently colored areas and placing a point (1) at a distance of x/2 along the perpendicular to the tangent to this boundary (assuming it is uneven). Next, we try to find another point of the same color as (1) along the entire circumference with a radius of x (taking (1) as the center of the circle). If we don't find such a point, the entire circumference coincides with (1) in color, then we place point (2) on that line, which is perpendicular to the tangent (and on which point (1) is already located), at a distance of x from (1), and then we move the segment along this perpendicular. Within the diameter of the circle 2x, there must be an even number of color changes in the plane, so the desired distance x must be found. Yes, if it suddenly turns out that it cannot be found because our plane is divided by color like concentric rings from a diffraction grating, and we have chosen (1) as the center of these rings, then we can always find 2 identical points on one of the chords of the rings. This essentially simplifies the answer. If there are no color matches with (1) along the circumference with a radius of x from (1), then there are 2 points of the same color on the circumference itself, and a chord can be constructed between them, the length of which is equal to the radius.