Question 23
Find all triples of positive integers $(a,b,c)$ with $a+b+c=10$ for which there exist $a$ red, $b$ blue and $c$ green points in the plane, all distinct, with these properties:
- The sum of distances between every red point and every blue point is $37$.
- The sum of distances between every green point and every red point is $30$.
- The sum of distances between every blue point and every green point is $1$.
- The sum of distances between every red point and every blue point is $37$.
- The sum of distances between every green point and every red point is $30$.
- The sum of distances between every blue point and every green point is $1$.