KöMaL 2015 B4732

The teacher of a class of 36 students enters the averages of the mathematics test grades in a $6 \times 6$ table. Each student has a different mean grade. The teacher marks the largest entry in each column of the table. He finds that all of the 6 marked numbers lie in different rows. Then he marks the largest entry in each row, and finds that these all lie in different columns. Prove that the two sets of six numbers are equal.