Two groups of students, Group X and Group Y, each containing 12 students, took the same math test. The average (arithmetic mean) score for Group X was 85 points, and the average score for Group Y was 80 points.

Which of the following statements about the test scores MUST be true?
I. The student with the highest score in Group Y scored lower than the student with the lowest score in Group X.
II. If a student is randomly selected from Group X and another from Group Y, the student from Group X will always have a higher score.
III. It is impossible for every student in Group X to have scored less than or equal to every student in Group Y.