An astronomer observed 20 stars with his telescope. When he added up all the pairwise distances between the stars, the result was X. Suddenly a cloud obscured 10 of the stars. Prove that the sum of the pairwise distances between the 10 re- maining stars is less than 1/2 X.
Bonus: Can you improve the bound? What is the smallest real number r such that the new sum is always less than rX, regardless of the configuration of the stars?